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THE WERNER 

Mental Arithmetic 



CONTAINING 



The Principles of Arithmetic 



NUMEROUS PROBLEMS FOR ORAL ANALYSIS 
AND SOLUTION 



Albert N. Raub, Ph.D. 

President of Delaware College, and Author of " School Manaqement,' 

"Methods of Teaching," A Practical Rhetoric," "Studies in 

English and American Literature." 




CHICAGO NEW YORK 

The Werner Company 



Q.AI02. 



R3 



Copyright, 1894 by ALBERT N. Raub 
Copyright 1894, by The Werner Company 



ac. 



PEEFAOE. 



The educational public have become so accustomed to the 
term mental arithmetic, and the title has become so distinc- 
tive, that it would probably be unwise to call the exercises 
prepared for a book of this kind by their proper name — oral 
arithmetic. Mental Arithmetic is therefore used as the name 
of the book, because this title has become most familiar to 
both teacher and pupil. 

The author has tried to make this work not only progres- 
sive and practical in its character, but also to adapt it to the 
wants of those who believe that oral and written arithmetic 
should be taught in conjunction, the mental or oral arithmetic 
being made to serve the purpose of developing the principles 
of the science by a class of problems simpler than are usually 
found in the written arithmetic, where the help of pen or 
pencil is an excuse for their presence. 

Much of the opposition which once existed against the 
study of mental arithmetic arose partly because the subject 
was made too difficult, and partly because there was no visible 
connection, as ordinarily taught, between the oral and the 
written process. In order to overcome these objections all 
ambiguous or mere puzzle problems have been discarded, and 



PREFACE. 



. special -effort^ h^ve';been made to develop the principles of 
'tile sdenci^'iti harmony with the order and methods found in 
works on arithmetic where both the oral and the written 
process are used, but giving a greater number and variety of 
problems for oral solution than could profitably be given in a 
book in which both processes are combined. 

A. N. R. 
Delaware College, Del. 



EDUCATION DEPT. 



MENTAL ARITHMETIC- 



ADDITION. 



1. A boy has 1 right eye and 1 left eye ; how many eyes 
has he ? 

2. If I have 2 apples in one hand and 1 apple in the 
other, how* many apples have I ? 

8. Mary has 2 books and her sister has 2 ; how many have 
they together ? 

4. Mary had 3 cents and found 1 ; how many had she 
then? 

5. A boy had 3 dollars and earned 2 dollars more ; how 
many dollars had he then ? 

6. James found 2 apples and George found 3 ; how many 
did they both find ? 

7. Tabby caught 8 mice and Tom caught 3 ; how many 
did they both catch? 

8. A pencil cost 4 cents, and some pens 3 cents ; how much 
did they both cost ? 

9. If a boy has 5 cents and his sister has 2 cents, how 
many cents have they together? 

10. How many cents are 6 cents and 1 cent ? 

11. A man earned 4 dollars on Monday and 4 dollars on 
Tuesday ; how much did he earn in the two days ? 

541822 5 



6 ' ;: : .; • : mental arithmetic. 

,, 12,.4 Jaanes bas'5*<?ents and Samuel has 3 cents; how many 
:\ ,'c^iits' H^vo the^' together ? 

13. How many are 6 cents and 2 cents ? 

14. How many are 7 cents and 1 cent ? 

15. Henry had 8 dollars, and earned 1 dollar more ; how 
many had he then ? 

16. If George has 7 apples and I give him 2 more, how 
many will he then have? 

17. Sarah baked 4 loaves of bread last week and 5 loaves 
this week ; how many loaves did she bake in the two weeks ? 

18. A boy paid 6 cents for paper and 3 cents for pens; 
how much did he pay for both? 

19. How many are 2 boys and 7 boys? 3 cents and 6 
cents? 4 apples and 5 apples? 1 orange and 8 oranges? 

20. Harry had 5 cents and earned 5 cents more; how 
many had he then ? 

21. There are 4 boys in one class and 6 in another ; how 
many are there in both classes ? 

22. A man has 3 horses and 7 cows ; how many animals 
has he? 

23. A boy has 8 turkeys and 2 hens ; how many fowls 
has he ? 

24. If a boy earns 1 dollar and his father earns 9 dollars, 
how much do they together earn ? 

25. How many are 9 dollars and 1 dollar ? 6 men and 4 
men ? 7 apples and 3 apples ? 8 days and 2 days ? 5 cents 
and 5 cents ? 

26*. How many are — 

1 and 1 ? 2 and 3 ? 2 and 4 ? 2 and 5 ? 

2 and 1 ? 1 and 4? 1 and 5? 6 and 1 ? 

3 and 1 ? 3 and 2? 4 and 2? 4 and 4? 
4andl? 3 and 3? 4 and 3? 3 and 5? 

* If the learner find difficulty with the abstract numbers, let him at 
first use counters. 



ADDITION. 

27. How many are — 

2 and 6? 4 and 5? 5 and 5? 8 and 2? 
7andl? 3 and 6? 6 and 4? 7 and 3? 
7 and 2 ? 8 and 1 ? 1 and 9 ? 4 and 6 ? 

28. How many are — 

7andl? 4 and 5? 6 and 2? 4 and 6? 

4 and 2? 7 and 2? 2 and 5? 4 and 5? 

3 and 3? 5 and 1 ? 3 and 4? 3 and 7? 



1. William had 10 cents and found 1 cent; how many cents 
had he then ? 

2. There are 10 eggs in one nest and 2 in another ; how 
many eggs in both nests ? 

3. A boy has 10 hens and 3 ducks; how many fowls has he? 

4. There are 10 boys in class and 4 girls ; how many pupils 
in the class ? 

5. A coat cost 10 dollars and a hat 5 dollars ; how much 
did both cost ? 

6. I had 11 dollars and earned 1 dollar more; how many 
dollars had I then ? 

7. A boy had 11 books and bought 2 more; how many 
books had he then ? 

8. If I pay 11 cents for a slate and 3 cents for pens, how 
much do I pay for both ? 

9. Mary gave me 11 plums and Henry gave me 4; how 
many plums have I ? 

10. Bought a slate for 12 cents and sold it for 1 cent more; 
how much did I get for it ? 

11. A farmer sold 12 sheep to one man and 2 to another ; 
how many sheep did he sell ? 

12. A pupil answered 12 questions correctly and 3 incor- 
rectly ; how many did he try to answer ? 



8 MENTAL ARITHMETIC. 

13. George is 13 years old, his sister is 1 year older ; how 
old is she ? 

14. James rode 13 miles and walked 2 ; how far did he 
travel ? 

15. Harry jumped 14 feet, but William jumped 1 foot 
farther; how far did William jump? 

16. How many are 2 cents and 10 cents ? 3 books and 10 
books ? 4 men and 10 men ? 5 horses and 10 horses ? 

17. How many are 1 boy and 11 boys? 2 peaches and 11 
peaches? 3 hats and 11 hats? 4 pens and 11 pens? 

18. How many are 1 pencil and 12 pencils? 3 dollars and 
12 dollars? 2 slates and 12 slates ? 

19. How many are 1 bell and 13 bells? 1 box and 14 
boxes? 2 pins and 13 pins? 

20. How many are 9 cents and 3 cents ? 9 cents and 4 
cents ? 9 cups and 5 cups ? 6 apples and 9 apples ? 

21. How many are 6 boys and 5 boys? 6 hens and 9 
hens ? 8 sheep and 6 sheep ? 7 men and 7 men ? 

22. How many are 7 plums and 5 plums ? 6 peaches and 

7 peaches ? 7 miles and 8 miles ? 

23. How many are 8 boys and 4 boys ? 6 men and 8 men ? 

8 cows and 5 cows ? 8 feet and 7 feet ? 

24. How many are — 



7 and 4? 


7 and 5? 


7 and 6? 


10 and 2? 


8 and 4? 


8 and 5? 


8 and 6? 


10 and 3? 


9 and 4? 


9 and 5 ? 


9 and 6? 


10 and 4? 


25. How many 


are — 






6 and 5 ? 


8 and 3 ? 


4 and 9? 


3 and 8? 


6 and 6? 


8 and 7? 


4 and 8? 


3 and 10? 


6 and 7? 


6 and 8? 


4 and 7 ? 


3 and 12? 


26. How many 


are — 






5 and 6? 


2 and 11? 


3 and 10? 


7 and 8? 


5 and 8 ? 


2 and 12? 


3 and 11? 


4 and 10? 


5 and 10? 


2 and 13? 


3 and 12? 


5 and 9? 



ADDITION. y 

3. 

1. A man had 15 dollars and earned 1 dollar more; how 
much had he then ? 

2. A coat cost 15 dollars and a hat 5 dollars ; how much did 
both cost ? 

3. Susan had 15 cents and her brother gave her 3 more ; 
how many had she then? 

4. James having 2 dollars earned 15 more ; how many 
dollars had he then? 

5. Samuel has 4 pears and John has 15 ; how many have 
they together ? 

6. A farmer having 16 cows bought 3 more ; how many 
had he then? 

7. James found 2 eggs in one nest and 16 in another ; how 
many eggs did he find ? 

8. Mary had 4 chickens, but now she has 1 6 more ; how 
many has she now? 

9. Mary is 17 years old and her brother is 2 years older ; 
how old is he ? 

10. Samuel had 17 chestnuts in his pocket and found 3 
more ; how many had he then ? 

11. If a boy is 18 years old, how old will he be in 2 years 
from now? 

12. How many are 2 men and 15 men ? 4 apples and 15 
apples? 3 cents and 15 cents? 

13. How many are 4 boys and 16 boys? 3 books and 16 
books? 2 pens and 16 pens? 

14. How many are 1 day and 17 days? 3 days and 17 
days? 17 weeks and 2 weeks? 

15. How many are 2 minutes and 18 minutes? 18 hours 
and 1 hour? 

16. How many are 10 cents and 10 cents? 8 dollars and 
8 dollars ? 10 flies and 8 flies ? 



10 MENTAL ARITHMETIC. 

17. How many are 9 men and 10 men ? 7 figs and 9 figs? 
6 balls and 12 balls? 

18. How many are 8 plates and 9 plates ? 7 plums and 9 
plums? 8 feet and 7 feet? 

19. How many are 9 hens and 5 hens? 10 pigs and 6 
pigs ? 14 sheep and 5 sheep ? 

20. How many are — 

6 and 11? 7 and 11? 5 and 13? 8 and 8? 
5 and 12? 8 and 9? 6 and 13? 9 and 8? 

7 and 9? 9 and 7? 7 and 13? 8 and 10? 
The sign +, called plus, when placed between two numbers indicates 

that they are to be added. Thus, 6 + 3 means 6 and 3. 

21. How many are — 

9+ 8? 9+ 7? 5 + 11? 4 + 16? 3 + 16? 
9+ 9? 4 + 12? 5 + 14? 4+14? 16+ 3? 
9 + 10? 4 + 13? 5 + 15? 4 + 15? 3 + 14? 

4. 

1. James has 4 dollars, William has 7, and Harry has 5 ; 
how many have they together ? 

Solution. — Since James has 4 dollars, William 7, and Harry 5, they 
together have 4 dollai*s, 7 dollars, and 5 dollai's, which is 16 dollars. 

2. A farmer has 7 acres of wheat, 5 acres of oats, and 3 
acres of barley ; how many acres of grain has he ? 

3. I paid 5 dollars for shoes, 6 dollars for a hat, and 4 dol- 
lars for an umbrella ; how much did I pay for all ? 

4. A boy worked 5 days in one week, 6 days the next, and 3 
days the next ; how many days did he work in the three weeks? 

5. I gave 5 cents to one boy, 6 to another, and 8 to another ; 
how many cents did I give to all ? 

6. I paid 10 dollars for coal, 8 dollars for wood, and 4 
dollars for hauling ; how much did my fuel cost ? 

7. George has 4 peaches, James has 12, and Henry 10 ; 
how many have they together? 



ADDITION. 11 

8. A lady paid 6 dollars for a hat, 5 dollars for shoes, and 
12 dollars for a dress-; how much did she pay for all? 

9. A barrel of flour cost 6 dollars, a barrel of sweet potatoes 
4 dollars, and a ton of coal 5 dollars ; how much did all cost ? 

10. James has 10 marbles, Henry has 8, and William has 
12 ; how many have they together? 

11. Samuel shot 8 birds, William 9, and Joseph 10; how 
many did they all shoot ? 

12. If a slate cost 12 cents and some paper 10 cents, how 
much do both cost ? 

13. A boy has 5 sisters and 4 brothers ; how many children 
are there in the family ? 

14. Robert is 16 years old and his sister is 7 years older ; 
how old is the sister ? 

15. There are 13 girls in a class and 12 boys ; how many 
pupils in the class ? 

16. If Mary has 24 hens and 6 ducks, how many fowls 
has she? 

17. A farmer has 12 cows, 6 horses, and 10 hogs; how 
many animals has he? 

18. A boy in searching for eggs found 12 in one nest, 10 
in another, and 8 in another ; how many eggs did he find ? 

19. Harry has 10 cents, James has 12, and John has 9; 
how many cents have they together? 

20. Lewis found 6 pears under one tree, 8 under another, 
and 13 under another, how many pears did he find ? 

21. Lucy has 7 roses, Laura has 10, and Mary has 14; how 
many roses have they together ? 

22. John is 10 years old, Annie is 4 years older, and Henry 
is 8 years older than Annie ; how old is each ? 

23. I gave 5 cents for a pencil, 12 cents for some paper, 
and 10 cents for a tablet ; how much did I pay for all ? 

24. My copy-book cost me 15 cents, my ink 10 cents, and 
my pens 6 cents ; how much did they all cost ? 



12 MENTAL ARITHMETIC. 

25. Bought a pound of sugar for 8 cents, a pound of rice 
for 9 cents, and a pound of coffee for 22 cents ; how much 
did I pay for all? 

5. 

1. There are 16 boys in school, and 15 girls; how many 
pupils in the school? 

2. James walked 15 miles in the forenoon and 12 miles in 
the afternoon ; how far did he walk ? 

3. Henry can earn 10 cents an hour and Samuel 16 cents 
an hour; how much can they together earn in an hour? 

4. A lady paid 15 cents for soap, 5 cents for pins, and 12 
cents for candy ; how much did she pay for all? 

5. A man sold 10 cords of wood one day, 8 cords the next, 
and 15 cords the next ; how many cords did he sell ? 

6. How many are 10 horses and 15 horses? 12 boys and 
8 boys? 20 cents and 12 cents? 

7. How many are 12 horses and 12 horses? 15 cows and 
12 cows? 15 sheep and 15 sheep? 

8. How many are 14 men and 10 men? 14 boys and 12 
boys? 14 cents and 15 cents? 

9. How many are 16 bushels and 14 bushels? 16 days 
and 12 days ? 10 weeks and 16 weeks? 

10. How many are 18 birds and 8 birds ? 10 ducks and 

18 ducks? 12 hens and 18 hens? 

11. How many are 17 pigs and 8 pigs? 10 robins and 17 
robins? 12 plums and 7 plums? 

12. How many are 20 hens and 10 hens? 8 pins and 20 
pins ? 20 cups and 9 cups ? 

13. How many are 19 dollars and 10 dollars? 8 cents and 

19 cents? 9 trees and 19 trees? 

14. How many are 21 roses and 7 roses? 6 pinks and 21 
pinks ? 21 chestnuts and 9 chestnuts ? 

15. How many are 8 oranges, 10 oranges, and 6 oranges? 



ADDITION. 13 

16. How many are 3 dollars, 5 dollars, and 12 dollars? 

17. How many are 6 boys, 8 boys, and 12 boys? 

18. How many are 10 sheep, 12 sheep, and 8 sheep? 

19. How many are 16 pigs, 7 pigs, and 6 pigs? 

20. How many are 8 doves, 10 doves, and 18 doves? 

21. How many are — 

14 + 10? 164-8? 18 + 10? 15+ 7? 17 + 4? 

14 + 12? 16 + 7? 14 + 16? 15 + 10? 17 + 9? 

16+ 5? 21 + 4? 16+ 9? 15+ 8? 16 + 5? 

22. How many are — 

14 + 13? 15 + 7? 16+ 6? 19+ 9? 22 + 7? 

13 + 10? 16 + 8? 17+ 7? 19+ 5? 20 + 9? 

10 + 18? 15 + 7? 18+ 8? 19+ 6? 21 + 6? 

23. How many are — 

6 + 3 + 4? 8+ 4+ 2? 5+ 6 + 7? 10+ 4+ 6? 

7 + 2 + 8? 9+ 3+ 5? 6+ 7 + 8? 8+ 3 + 10? 

6 + 5 + 3? 6+ 9+ 4? 7+ 8 + 9? 9+ 7 + 10? 

24. How many are — 

7 + 6 + 3? 8+ 5+ 6? 10+ 3 + 7? 11+ 3+ 4? 
6 + 8 + 2? 5+ 7 + 10? 9+ 2 + 6? 11+ 6+ 2? 

8 + 9 + 3? 7 + 10+ 9? 8+ 1 + 9? 11+ 5+ 7? 

25. How many are — 

12 + 4 + 6? 14 + 6+ 2? 16 + 10 + 4? 18+ 2+ 8? 
12 + 6 + 8? 14 + 4+ 4? 14+ 6 + S? 4 + 19+ 6? 
12 + 3 + 5? 14 + 8+ 6? 15+ 7 + 3? 8+ 2 + 17? 

26. Count by 2's from 2 to 50. Thus, 2, 4, 6, 8, 10, etc. 

27. Count by 5's from 5 to 50 ; from 50 to 80. 

28. Count by 3's from 3 to 60 ; from 60 to 99. 

29. Count by 4's from 1 to 49 ; from 50 to 98. 

30. Count by 6's from 2 to 50 ; from 50 to 90. 

31. Declare the sums of the following columns : 
78628347845434894 
37794664667268768 
4, 4, 3, 9, 8, 9, 8, 7, 8, 9, 3, 6, 9, 5, 5, 9, 6. 



14 MENTAL ARITHMETIC. 

49899939666672443 
48735674478888754 

3, 9, 8, 6, 5, 4, 7, 6, 6, 2, 7, 8, 6, 3, 8, 4, 8. 

18546768814613738 
36472838786846286 
9, 2, 3, 6, 5, 6, 7, 4, 9, 5, 7, 8, 6, 4, 8, 7, 8. 

38296756898769986 
64625474799549997 

4, 6, 8, 3, 6, 9, 8, 8, 8, 6, 4, 8, 9, 8, 9, 7, 8. 

73388258484778494 
31679867336756686 
43448478268678879 
4, 9, 7, 9, 6, 6, 3, 5, 6, 4, 3, 3, 8, 6, 7, 8, 6. 

54884849574487448 
92878396488858774 
96655399396874568 
6, 6, 8, 8, 2, 6, 8, 4, 8, 7, 3, 6, 3, 9, 8, 7, 5. 

74885754648677524 
43468498889975763 
34814785678788944 
39894487523289186 
3, 6, 2, 2, 9, 8, 9, 7, 3, 8, 7, 5, 8, 3, 3, 6, 8. 

12 14 11 18 16 27 14 18 16 21 18 16 

16 11 24 22 20 11 30 17 25 32 26 24 

10 12 16 25 13 13 42 15 40 45 20 34 

12, 13, 10, 15, 12, 26, 16, 14, 17, 12, 14, 41. 



-SUBTRACTION. 15 

SUBTRACTION. 
6. 

1. John had 3 cents and lost 1 of them ; how many cents 
has he remaining? 

Solution. — Since John had 3 cents and lost 1 of them, he has re- 
maining the difference between 3 cents and 1 cent, which is 2 cents. 

2. A boy had 4 apples and ate 2 ; how many has he re- 
maining ? 

3. If Susan has 5 dollars and spends 3 dollars for a hat, 
how much money will she have left ? 

4. 6 birds sat on a tree, but 3 have flown away ; how many 
remain ? 

5. A boy had 7 pins, but lost 4 ; how many has he re- 
maining ? 

6. Henry is 6 years old, but his brother is 2 years younger ; 
how old is the brother ? 

7. A boy had 7 eggs, but he broke 3 ; how many remain ? 

8. Susan is now 8 years old ; how old was she 3 years 
ago? 

9. There are 7 pupils in the class, of whom 2 are girls ; 
how many boys in the class ? 

10. If a boy have 10 cents, and spend 5, how many cents 
will remain ? 

11. How many are 8 cents less 3 cents? 6 buttons less 1 
button ? 9 dollars less 4 dollars ? 

12. How many are 6 oranges less 2 oranges? 8 dolls less 
5 dolls ? 7 books less 5 books ? 

13. How many are 7 boys less 3 boys? 6 pens less 2 pens? 
8 peaches less 6 peaches ? 

14. How many are 8 apples less 4 apples ? 10 pears less 6 
pears? 7 pencils less 4 pencils? 



16 MENTAL ARITHMETIC. 



15. How many 


are — 






4 less 1 ? 


5 less 2? 


7 less 2? 


10 less 7? 


4 less 3? 


5 less 4? 


9 less 6? 


9 less 4? 


4 less 2? 


5 less 3? 


9 less 3? 


8 less 5? 


16. How many 


are — 






10 less 5? 


6 less 3? 


7 less 4? 


8 less 3? 


10 less 8? 


6 less 4? 


7 less 6? 


8 less 4? 


10 less 6? 


6 less 2 ? 


7 less 2? 


8 less 6? 



7. 

1. Mr. A had 20 sheep, but sold 10 ; how many had he 

remaining ? 

2. Bought a slate for 16 cents and sold it for 12 cents ; how 
much did I lose ? 

3. If a man buy a sheep for 10 dollars and sell it for 15 
dollars, how much will he gain ? 

4. A class has 18 pupils, but 4 are absent ; how many are 
present ? 

5. James is 16 years of age and Harry is 13 ; how much 
older is James than Harry ? 

6. A boy had 20 quarts of nuts and sold 6 quarts of them ; 
how many quarts remain ? 

7. Samuel had a flock of 17 hens, but 6 have died ; how 
many remain ? 

8. A bush had 16 roses, but 10 of them have faded ; how 
many remain ? 

9. A man planted 18 trees, but 7 have died ; how many 
are living? 

10. If a man had 14 bushels of beans and sold 8 bushels, 
how many bushels remain ? 

11. If a man buy a colt for 10 dollars and sell it for 16 
dollars, how much does he gain? 

12. A girl who had 20 cents bought a pound of sugar for 
8 cents ; how much money had she remaining ? 



SUBTRACTION. 17 

13. Samuel gathered 13 quarts of berries and sold 4 quarts; 
how many did he keep ? 

14. A farmer has a flock of 19 turkeys ; how many will he 
have remaining if he sell 14 of them? 

15. How many are 16 boys less 4 boys? 15 pens less 8 
pens? 17 pencils less 6 pencils? 

16. How many are 20 men less 8 men ? 14 roses less 11 
roses? 20 ducks less 12 ducks? 

17. How many are 19 sheep less 15 sheep? 16 oranges 
less 7 oranges? 18 cents less 9 cents? 

18. How many are 19 years less 9 years? 20 days less 11 
days? 15 hours less 9 hours? 

19. How many are 19 tops less 5 tops? 14 marbles less 8 
marbles? 18 toys less 6 toys? 

20. How many are — 



16 less 4? 


14 less 6? 


13 less 5? 


12 less 7? 


18 less 6? 


15 less 5? 


11 less 7? 


13 less 6? 


20 less 2? 


16 less 8? 


12 less 8? 


11 less 8? 


21. How many 


are — 






15 less 6? 


16 less 9? 


18 less 5? 


19 less 7? 


15 less 12? 


17 less 8? 


18 less 10? 


20 less 3? 


16 less 6? 


17 less 5 ? 


19 less 15? 


20 less 9? 



Note. — The sign for subtraction is a short horizontal line, called 
minus. When placed between two numbers it means that the second 
is to be subtracted from the first. Thus, 8 — 2 means that 2 is to be 
taken from 8. 



22. How many are — 






11-3? 16- 4? 


20-17? 


19- 6? 


12-6? 18-12? 


15-4? 


17-14? 


13-5? 16- 7? 


18-11? 


14- 9? 



8. 

1. Mary had 25 cents and gave 5 cents for a pencil ; how 
much money has she remaining ? 
2 



18 MENTAL ARITHMETIC. 

2. A farmer bought a cow for 20 dollars and sold her for 30 
dollars ; how much did he gain ? 

3. If a man earn 24 dollars a week and pay 6 dollars a 
week for board, how much does he save ? 

4. There are 28 pupils in a school, of whom 18 are girls ; 
how many boys are there ? 

5. A farmer having 25 gallons of vinegar sold 12 gallons ; 
how many gallons has he remaining ? 

6. Mr. Jones bought a cow for 23 dollars and sold her for 
28 dollars ; how much did he gain ? 

7. From a school of 30 pupils 9 were promoted ; how many 
remained ? 

8. A farmer sold 20 dollars' worth of hay, and received in 
exchange 12 dollars' worth of coal and the remainder in cash; 
how much cash did he get ? 

9. Charles had 29 chickens, but 12 have been sold; how 
many remain ? 

10. William is 24 years old and his sister is 8 years 
younger; how old is she? 

11. A lady who earns 40 dollars a month pays 20 dollars a 
month for board ; how much does she save ? 

12. If I had 36 cents and spent 14 cents for a slate, how 
much had I remaining? 

13. Susan had 45 cents and Mary 32 cents; how much 
more had Susan than Mary? 

14. If a slate cost me 12 cents, how much do I gain by 
selling it for 23 cents? 

15. John is 15 years old; how long before he will be 21 
years of age? 

16. A boy started to walk to a city 35 miles away; he 
walked 16 miles the first day ; how far did he walk the 
second day to reach the city? 

17. Bought a wagon for 45 dollars and sold it for 28 dol- 
lars ; how much did I lose ? 



ADDITION AND SUBTRACTION. 19 

18. A man had 43 fowls, but after a fox had caught some 
he had only 'SQ ; how many did the fox catch ? 

19. I am now 48 years old ; how old was I 15 years ago? 

20. If you are now 14 years old ; how long before you will 
be 40 years of age ? 

21. How many are 46 dollars less 18 dollars? 27 cents 
less 12 cents? 43 books less 26 books? 

22. How many are 34 apples less 18 apples? 48 pages less 
16 pages? 35 years less 16 years ? 

23. How many are 38 chestnuts less 17 chestnuts? 47 
pupils less 18 pupils? 

24. How many are 50 miles less 16 miles? 33 feet less 17 
feet? 

25. How many are 44 days less 32 days ? 35 dollars less 
22 dollars? 

26. How many are — 

42-^16? 50-18? 48-18? 33- 8? 

34-17? 45-26? 40-19? 47-10? 

27- 6? 39-12? 32-15? 43-13? 

27. How many are — 

38-14? 43-22? 41-25? 27-17? 
36-29? 30-18? 36- 8? 42-12? 
48-12? 47-17? 25-13? 38-27? 

28. Count by 2's from 50 to 2 ; from 100 to 50. 

29. Count by 5's from 100 to 60 ; from 60 to 20. 

30. Count by 4's from 100 to 40 ; from 40 to 0. 

31. Count by 6's from 90 to 60 ; from 60 to 12. 



ADDITION AND SUBTRACTION. 

1. Mary had 4 dollars and earned 10 more ; she then spent 
7 dollars ; how much money had she remaining ? 

2. Susan gathered 10 quarts of berries in the forenoon and 



20 MENTAL ARITHMETIC. 

15 quarts in the afternoon ; if she sold 16 quarts ; how many 
had she remaining ? 

3. A man having 16 cents gave 5 cents to one boy and 6 
to another ; how many cents had he remaining ? 

4. A farmer bought 12 sheep from one man and 16 from 
another; he then sold 20 of them; how many had he re- 
maining? 

5. Bought a pencil for 5 cents and a slate for 12 cents; 
how much change did I get by paying with a 25-cent piece ? 

6. Bought a cow for 30 dollars and a calf for 4 dollars ; I 
sold the two for 38 dollars ; how much did I gain ? 

7. A boy spent 18 cents for oranges and 15 cents for nuts ; 
how much more would he have to spend to make the amount 
50 cents ? 

8. A laborer having 20 dollars earned 15 dollars more, 
and then bought a suit of clothes for 18 dollars ; how much 
money has he remaining? 

9. Henry having 25 apples sold 8 and ate 6 ; how many 
has he remaining ? 

10. A man bought goods for 27 dollars and paid 3 dollars 
for hauling ; how must he sell the goods to gain 6 dollars ? 

11. George spent 15 cents, then earned 12 cents, and had 
17 cents ; how much had he at first ? 

12. Henry wishes to buy a book worth 50 cents : his father 
gave him 15 cents and his mother 12; how much must he 
earn to buy the book? 

13. Three boys have 48 cents : the first has 12 cents, and 
the second 19 cents ; how much has the third ? 

14. A boy got 12 peaches from one tree, 15 from another, 
and 10 from another ; he ate 6, gave away 14, and took the 
others home ; how many did he take home ? 

15. A gentleman bought a watch for 60 dollars and a chain 
for 12 dollars : he sold the two for 80 dollars ; how much did 
he gain ? 



ADDITION AND SUBTRACTION. 21 

16. A merchant had a piece of cloth containing 54 yards: 
he sold 12 yards to one man, 15 to another, and 10 to another; 
how many yards remain in the piece ? 

17. A man's wages for the month were 46 dollars : he paid 
6 dollars for groceries, 10 dollars for rent, and 12 dollars for 
meat ; how much had he remaining ? 

18. If a boy having 60 cents, spends 24 cents, and then 
earns 43 cents; how much will he have? 

19. Samuel is 24 years of age, which is 6 years more than 
Mary's age and 13 years more than Albert's age; how old is 
Mary and how old is Albert ? 

20. Two boys gathered some berries : the one gathered 24 
quarts and the other gathered 5 quarts less ; how many quarts 
did both gather? 

21. James exchanged eggs worth 25 cents and butter worth 
56 cents for a knife worth 90 cents ; how much cash did he 
have to pay ? 

22. Bought a horse for 120 dollars and a sleigh for 60 dol- 
lars ; how much did I gain by selling both for 185 dollars? 

23. A man having a certain number of sheep bought 20, then 
sold 40, and had 5 remaining ; how many had he at first? 

24. Harry and James began playing with 25 marbles each : 
at the end of the game James had 32 ; how many had Harry? 

25. Mr. A bought a cart for 30 dollars, some harness for 
15 dollars, and a whip for 3 dollars : he sold all for 16 dollars 
more than the cart cost ; how much did he lose? 

26. Mr. Smith walked 25 miles one day and 20 miles the 
next : on the third day he returned 32 miles ; how far was he 
then from the starting-place ? 

27. A farmer had 60 sheep in one field and 54 in another : 
he sold 25 from each field ; how many remained in each ? 

28. A man bought a cow for 45 dollars, some sheep for 18 
dollars, and pigs for 25 dollars : he exchanged them all for a 
horse worth 100 dollars'; how much cash did he have to pay? 



22 



MENTAL ARITHMETIC. 



Find the value — 






29. Of 4 + 6 - 3 




37. Of 20 + 25 - 15 


30. Of 8 + 7 - 6 




38. Of 36 + 14- 12 


31. Of 10 + 12- 9 




39. Of 30 - 6 - 4 


32. Of 15+ 5- 8 




40. Of 40 - 18 + 6 


33. Of 18 + 20 - 16 




41. Of 60-24 + 21 


34. Of 25 + 18 - 14 




42. Of 22 - 14 + 63 


35. Of 20 + 10 - 6 




43. Of 70 - 17 + 7 


36. Of 22 + 18 - 10 




44. Of 75 - 34 + 22. 


45. How many are 6 


plus 4 


[ minus 3 ? 


46. How many are 2C 


) minus 16 plus 8? 


47. How many are 32 


I plus 


18 minus 25 ? 


48. How many are 4^ 


> minus 16 plus 22? 


49. How many are 6C 


) plus 


15 minus 56 ? 


50. How many are 36 


> minus 27 plus 33 ? 



MULTIPLICATION. 

1. How much will 2 oranges cost at 3 cents each? 

Kemark. — Since 1 orange costs 3 cents, two oranges will cost 3 cents 
and 3 cents, or 2 times 3 cents, which is 6 cents. 

Solution. — Since one orange costs 3 cents, 2 oranges will cost 2 times 
3 cents, or 6 cents. 

2. How much will 2 pencils cost at 4 cents each ? 

3. How many are 2 times 2 boys ? 2 times 4 pins ? 

4. How many are 2 times 5 days? 2 times 6 apples? 

5. How many are 2 times 7 weeks ? 2 times 10 cents ? 

6. How many are 2 times 8 sheep ? 2 times 9 peaches ? 

7. How many are — 

2 times 1 ? 2 times 4 ? 2 times 7 ? 2 times 10 ? 

2 times 2 ? 2 times 5 ? 2 times 8 ? 2 times 11 ? 

2 times 3 ? 2 times 6 ? 2 times 9 ? 2 times 12 ? 
. Remark. — 2 times 3 is the same as 2 threes. 



MULTIPLICATION. 23 

8. If a lemon cost 3 cents, how much will 3 lemons cost ? 

9. At 4 cents each how much will 3 pencils cost ? 

10. How much will 3 oranges cost at 5 cents each ? 

11. At 6 cents each how much will 3 bottles of ink cost? 

12. How many are 3 times 7 roses? 3 times 8 dollars? 

13. How many are 3 times 9 minutes? 3 times 10 hours? 

14. How many are 3 times 11 days? 3 times 12 eggs? 

15. How many are — 

3 times 1 ? 3 times 4 ? 3 times 7 ? 3 times 10 ? 
3 times 2 ? 3 times 5 ? 3 times 8 ? 3 times 11 ? 
3 times 3 ? 3 times 6 ? 3 times 9 ? 3 times 12 ? 

16. How much will 4 apples cost at 2 cents each ? 

17. At 4 cents a pint how much will 4 pints of milk cost? 

18. At 6 cents a quart what will 4 quarts of beans cost? 

19. At 10 cents each what will 4 slates cost ? 

20. How many are 4 times 3 cents ? 4 times 5 cents ? 4 
times 7 cents ? 

21. How many are 4 times 8 sheep? 4 times 9 hens? 

22. How many are 4 times 11 roses? 4 times 12 pinks? 

23. How many are — 

4. ones ? 4 fours ? 4 sevens ? 4 times 10 ? 
. 4 twos ? 4 fives ? 4 eights ? 4 times 11 ? 
4 threes? 4 sixes? 4 nines? 4 times 12? 

Note. — The sign of multiplication is X j which is read times. Thus, 
4 X 3 is read 4 times 3. 

24. If 1 hat cost 2 dollars, how much will 5 hats cost ? 

25. If a pencil cost 5 cents, how much will 5 pencils cost ? 

26. How much will 5 sheep cost at 8 dollars each ? 

27. There are 7 days in a week; how many days in 5 
weeks ? 

28. How many are 5 times 3 apples? 5 times 4 apples? 
, 29. How many are 5 times 6 pears ? 5 times 9 pears ? 

30. How many are 5 times 10 hens ? 5 times 11 hens? 5 
times 12 hours ? 



24 MENTAL ARITHMETIC. 

31. How many are — 

5 ones? 5times4? 5 times 7? 5X10? 
5 twos ? 5 times 5 ? 5 times 8 ? 5X11? 
5 threes ? 5 times 6 ? 5 times 9 ? 5X12? 

32. There are 3 feet in a yard ; how many feet in 6 yards? 

33. At 5 cents a yard how much will 6 yards of calico cost ? 

34. If a boy earns 8 dollars a week, how much does he 
earn in 6 weeks ? 

35. How many are 6 times 2 boys? 6 times 4 boys ? 

36. How many are 6 times 6 books ? 6 times 7 books ? 

37. How many are 6 times 9 feet ? 6 times 10 feet? 

38. How many are 6 times 11 inches? 6 times 12 inches? 

39. How many are — 

Six I's ? 6 times 4 ? 6 times 7 ? 6X10? 
Six 2's ? 6 times 5 ? 6 times 8 ? 6X11? 
Six 3's ? 6 times 6 ? 6 times 9 ? 6X12? 

40. How much will 7 bananas cost at 3 cents each ? 

41. How much will 7 tablets cost at 10 cents each? 

42. There are 7 days in a week ; how many days in 7 weeks ? 

43. How many are 7 times 2 plums? 7 times 8 plums? 

44. How many are 7 times 4 cups? 7 times 9 cups ? 

45. How many are 7 times 5 cents? 7 times 11 cents ? 

46. How many are 7 times 6 flowers? 7 times 12 flowers? 

47. How many are — 

7 times 1? 7 times 4? 7X7? 7X10? 
7 times 2? 7 times 5? 7X8? 7X11? 
7 times 3? 7 times 6? 7X9? 7X12? 

48. At 5 cents each how much will 8 pencils cost? 

49. A man earns 10 dollars a week; how much does he 
earn in 8 weeks? 

50. How many shoes in 8 pairs ? 

51. How many are 8 times 3 lemons? 8 times 4 lemons? 

52. How many are 8 times 6 boys? 8 times 7 boys? 

53. How many are 8 times 9 girls ? 8 times 10 girls ? 



MULTIPLICATION. 25 

54. How many are 8 times 11 men? 8 times 12 men? 

55. How many are — 

8 times 1? 8X4? 8 times 7? 8X10? 
8 times 2? 8X5? 8 times 8? 8X11? 

8 times 3? 8XQ? 8 times 9? 8X12? 

56. At 2 cents each how much will 9 peaches cost ? 

57. If a boy walk 3 miles an hour, how far can he walk 
in 9 hours? 

58. What is the cost of 9 pencils at 5 cents each ? 

59. How many are 9 times 4 acres ? 9 times 6 miles ? 9 
times 7 days? 

60. How many are 9 times 8 men ? 9 times 10 cents ? 9 
times 12 eggs? 

61. How many are 9 times 9 boys? 9 times 11 dolls? 

62. How many are — 

9timesl? 9X4? 9X7? 9timesl0? 

9 times 2? 9X5? 9X8? 9 times 11? 
9 times 3? 9X6? 9X9? 9 times 12? 

63. A chair has 4 legs ; how many legs have 10 chairs ? 

64. At 8 cents each how much will 10 books cost? 

65. There are 12 eggs in a dozen ; how many eggs in 10 
dozen ? 

6Q. How many are 10 times 2 cents? 10 times 5 cents? 
10 times 3 cents? 

67. How many are 10 times 6 loaves? 10 times 7 beans? 
10 times 8 birds? 

68. How many are 10 times 11 marbles? 10 times 12 
marbles ? 

69. How many are — 

10X1? 10 times 4? 10 times 7? 10X10? 
10X2? 10 times 5? 10 times 8? 10X11? 
10X3? 10 times 6? 10 times 9? 10X12? 

70. How many gloves in 11 pairs? 

71. How much can a man earn in 11 days at 5 dollars a day? 



26 MENTAL ARITHMETIC. 

72. How much will 11 tablets cost at 9 cents each? 

73. How many are 11 times 3 books? 11 times 6 books? 
11 times 4 books? 

74. How many are 11 times 7 pens? 11 times 9 pens? 

75. How many are 11 times 8 roses? 11 times 10 pinks? 

11 times 12 daisies? 

76. How many are — 

11X1? 11X4? 11 times 7? 11 times 10? 
11X2? 11X5? 11 times 8? 11 times 11? 
11X3? 11X6? 11 times 9? 11 times 12? 

77. How much will 12 hats cost at 3 dollars each ? 

78. How much can a man earn in 12 weeks at the rate of 
10 dollars a week ? 

79. How many days in 12 weeks? 

80. How many are 12 times 2 pens? 12 times 4 pens? 

12 times 5 books? 12 times 6 pages? 

81. How many are 12 times 8 pages? 12 times 11 words? 
12 times 9 feet ? 12 times 12 inches ? 

82. How many are — 

12X1? 12X4? 12X7? 12X10? 
12X2? 12X5? 12X8? 12X11? 
12X3? 12X6? 12X9? 12X12? 



2. 

1. How much will 3 cows cost at 30 dollars each? 

2. What is the cost of 5 bushels of potatoes at 40 cents a 
bushel ? 

3. If a train run 25 miles an hour, how far will it run in 
6 hours ? 

4. What is the cost of 5 pounds of coffee at 32 cents a 
pound ? 

5. What is the cost of 20 pounds of sugar at 8 cents a 
pound ? 



MULTIPLICATION. 27 

6. At 25 dollars each how much will 8 cows cost ? 

7. At 50 cents each how much will 4 books cost ? 

8. Henry is 15 years old, his father is 3 times as old ; how 
old is his father ? 

9. What is the cost of 33 yards of cloth at 3 dollars a yard? 

10. Susan bought 10 yards of gingham at 16 cents a yard ; 
how much did it cost ? 

11. How much will 15 yards of gingham cost at 16 cents 
a yard ? 

12. If a quire of paper cost 22 cents, how much will 3 
quires cost? 4 quires? 5 quires? 

13. If 6 men can do a piece of work in a day, how long 
will it take 1 man to do it ? 

14. If 5 men can do a piece of work in 3 days, how long 
will it take 1 man ? 

15. If 10 men can earn 90 dollars in a week, how long will 
it take 1 man to earn the same amount ? 

16. If 6 men dp a piece of work in 1 day, how long will it 
take 1 man to do the work ? 

17. Three men can plow a field in 4 days ; how many men 
will it require to plow the field in 1 day ? 

18. If a barrel of flour last 8 persons 6 weeks, how many 
persons will it last 1 week? 

19. If 12 men can do a piece of work in 10 days, how 
many men will it require to do it in 1 day? 

20. If 12 men can do a piece of work in 10 days, how long 
will it take 1 man to do the work ? 

21. What is the cost of 2 oranges at 5 cents each and 4 
bananas at 3 cents each ? 

22. A farmer sold 4 sheep at 5 dollars each, and 6 lambs 
at 3 dollars each ; how much did he get for all? 

23. Bought 2 slates at 15 cents apiece, and 3 pencils at 4 
cents each ; how much did I pay for all ? 



28 MENTAL ARITHMETIC. 

24. A farmer has 20 bags of wheat, each containing 3 
bushels; how much is it worth at 2 dollars a bushel? 

25. A man has 6 baskets, each containing 5 quarts of ber- 
ries ; how much are they all worth at 10 cents a quart ? 

26. A farmer sold 5 cows at 30 dollars each and 3 horses 
at 120 dollars each ; how much did he get for all ? 

27. A lady bought 2 pounds of rice at 8 cents a pound, 3 
pounds of sugar at 7 cents a pound, and 2 quarts of oil at 15 
cents a quart ; how much did she pay for all ? 

28. How much must I pay for 2 books at 30 cents each 
and 3 at 25 cents each ? 

29. A man and his son worked 12 hours: the father re- 
ceived 15 cents an hour and the son 10 cents an hour ; how 
much did they both receive ? 

30. A man bought 10 yards of cloth for a suit of clothes 
at 2 dollars a yard : he paid 6 dollars for making the coat, 2 
dollars for making the pantaloons, and 1 dollar for making 
the vest ; how much did the suit cost him ? 

31. George bought 3 pencils for 10 cents, and sold them at 
5 cents each ; how much did he gain? 

32. A boy bought a dozen bananas for 20 cents, and sold 
them at 3 cents each ; how much did he gain ? 

33. William earned 90 cents a day ; his board cost him 50 
cents a day ; how much did he save in 12 days? 

34. A drover bought 2 cows at 30 dollars each and sold 
the two for 6^ dollars ; how much did he gain ? 

35. If a man earn 16 dollars a week and pay 5 dollars a 
week for his board, how much can he save in 9 weeks ? 

36. James drives 6 miles an hour and Henry walks 3 miles 
an hour in the same direction ; how much farther does James 
travel than Henry in 12 hours? 

37. Samuel bought 6 pencils at 4 cents apiece and 5 tablets 
at 8 cents apiece ; how much did he gain by selling the lot 
for 80 cents? 



MULTIPLICATION. 29 

38. Henry has 10 hens and George has 3 times as many, 
less 4 ; how many have they together ? 

39. William has 15 cents, and Harry has 4 times as many, 
plus 8 cents ; how many has Harry ? 

40. A farmer bought 2 cows at 30 dollars each and 12 
sheep at 5 dollars each : he sold the lot for 150 dollars ; how 
much did he gain ? 

41. I had 6 cents and found 2 more ; how much must I 
earn to have 15 cents? 

42. Henry has 15 cents and David has 20 ; how much must 
they earn to have 60 cents together ? 

43. A and B start at the same point and travel in opposite 
directions. A at the rate of 3 miles an hour and B at the rate 
of 5 miles an hour ; how far apart are they at the end of 5 
hours? At the end of 6 hours? 10 hours? 

44. If A and B start at the same point and travel in the 
same direction, A at the rate of 4 miles an hour and B at the 
rate of 7 miles an hour, how far apart will they be in 3 hours? 
In 5 hours ? In 8 hours ? 

45. A farmer sells 5 pounds of butter at 25 cents a pound, 
and takes in exchange 20 pounds of sugar at 7 cents a pound ; 
how much cash does he have to pay? 

46. If a lady exchange 10 dozen of eggs at 15 cents a 
dozen for 15 pounds of sugar at 8 cents a pound, how much 
is still due her ? 

47. A drover bought 40 sheep for 110 dollars: he sold 10 
of them at 5 dollars a head, and the remainder at 3 dollars a 
head ; how much did he gain ? 

48. A lady bought 5 yards of silk at 2 dollars a yard, and 
6 yards of velvet at 3 dollars a yard : she gave the mer- 
chant 3 ten-dollar bills ; how much change did she get? 

49. There are 4 quarts in a gallon and 2 pints in a quart ; 
how many pints in 5 gallons ? 



30 MENTAL ARITHMETIC. 

50. James gathered 6 quarts of nuts, William gathered 3 
times as many, less 4 quarts, and Simon gathered as many as 
the other two ; how many did they all gather ? 

51. How many are 5 times 6, plus 16 ? 3 times 20, minus 
15? 

52. How many are 3 times 18, minus 6 ? 4 times 8, plus 
60? 

53. How many are 6 X 4, + 7 ? 16 X 5, + 18 ? 

54. How many are 7 X 15, - 12 ? 8 X 20, - 16 ? 

55. How many are 8 X 5, + 30 ? 8 X 10, - 23 ? 

56. How many are 8 X 12, + 15 X 2? 

57. How many are 10 X 15, — 6 X 7 ? 

58. How many are 11X6, + 14, — 25 ? 

59. How many are 18 X 5, + 15, + 10, - 16 ? 

60. How many are 10 + 5, X 6, — 25 ? 

Note.— 6 + 4X3 — 16 + 5X8 = 6, + 4X3, — 16, + 5X8 = 6 + 
TX3 — 16 + 5X8 = 6 + 12 — 16 + 40 = 42. 

When examples like the foregoing contain the signs +, — , and X, 
the numbers between which the multiplication sign, Xj is placed should 
be multiplied before the additions or subtractions are made. 



DIVISION. 



1. 

1. How many times are 2 cents contained in 6 cents ? 
Solution. — 2 cents are contained 3 times in 6 cents, because 3 times 

2 cents are 6 cents. 

2. How many times are 3 dollars contained in 6 dollars ? 

3. How many times are 4 pens contained in 12 pens ? 

4. How many times are 5 apples contained in 10 apples? 

5. How many times are 2 apples contained in 10 apples? 

6. How many times are 3 books contained in 12 books? 

7. How many times are 4 books contained in 12 books? 



DIVISION. 31 

8. How many times are 6 pencils contained in 18 pencils? 

9. How many apples at 2 cents each can I buy for 8 
cents ? 

Solution. — At 2 cents each I can buy as many apples for 8 cents a^ 
2 cents are contained times in 8 cents, or 4 apples. 

10. At 3 cents each how many oranges can I get for 15 
cents? 

11. At 4 dollars each how many hats can be bought for 20 
dollars ? 

12. At 4 cents each how many bananas can be bought for 
12 cents? 

13. If I divide 25 cents among 5 boys, how many cents 
does each boy get ? 

14. How many yards of silk at 4 dollars a yard can I get 
for 32 dollars? 

15. How many yards of tape at 3 cents a yard can I buy 
for 21 cents? 

16. How many weeks are there in 42 days? 

17. When bananas are 5 cents apiece, how many can I buy 
for 40 cents ? 

18. Some boys gather 30 quarts of nuts : each has 6 quarts ; 
how many boys are there ? 

19. A man spent 20 dollars for cloth at 5 dollars a yard ; 
how many yards did he buy ? 

20. At 3 dollars a barrel how many barrels of apples can 
be bought for 24 dollars ? 

21. If a pair of boots cost 8 dollars, how many pairs can 
be bought for 40 dollars ? 

22. How many barrels of flour at 7 dollars a barrel can 
be bought for 49 dollars? 

23. How many pounds of rice at 6 cents a pound can be 
bought for 60 cents ? 

24. If a man drive at the rate of 6 miles an hour, how 
long will it take him to drive 42 miles? 



32 MENTAL ARITHMETIC. 

25. A boat travels at the rate of 9 miles an hour ; how 
long will it take to go 54 miles ? 

26. If a boy can earn 10 cents an hour, how long will it 
take him to earn 90 cents ? 

27. When coal is worth 6 dollars a ton, how many tons can 
be bought for 48 dollars ? 

28. A farmer paid 45 dollars for sheep at 9 dollars apiece ; 
how many did he buy ? 

29. How many bars of soap at 10 cents a bar can I buy 
for 80 cents? 

80. How many ten-cent pieces are equal to 70 cents ? 

31. When hay sells at 11 dollars a ton, how many tons can 
be bought for 55 dollars ? 

32. At 11 cents a pound how many pounds of cheese can 
I buy for 88 cents? 

33. George has gathered 120 eggs ; how many dozen has 
he gathered ? 

34. How many slates at 12 cents each can I buy for 96 
cents ? 

35. How many hours at 11 cents an hour must I work to 
earn 66 cents? 

36. At 12 cents a quire how many quires of paper can I 
buy for 84 cents ? 

37. At 12 dollars a week how long will it take a man to 
earn 132 dollars? 

38. A farmer traded a horse worth 120 dollars for hogs 
worth 10 dollars each ; how many hogs did he get ? 

39. A dealer gave coal worth 6 dollars a ton for a load of 
wheat worth 72 dollars; how many tons of coal did it take? 

40. How many men at 9 dollars a week can be hired for 
72 dollars? 

41. Six roses are how many times 2 roses? 3 roses? 

42. 12 cents are how many times 6 cents? 3 cents? 4 
cents ? 



DIVISION. 33 

43. 15 cups are how many times 5 cups? 3 cups ? 

44. 18 dolls are how many times 3 dolls ? 9 dolls ? 

45. 24 men are how many times 3 men ? 6 men ? 4 men? 

8 men? 12 men? 

46. 30 books are how many times 5 books ? 10 books ? 6 
books ? 

47. 36 dollars are how many times 3 dollars ? 6 dollars ? 

9 dollars ? 12 dollars ? 18 dollars ? 

48. 48 marbles are how many times 4 marbles? 6 mar- 
bles ? 8 marbles ? 1 2 marbles ? 

49. 60 books are how many times 10 books? 5 books? 
12 books? 6 books? 20 books? 

50. 72 cents are how many times 8 cents? 12 cents? 9 
cents ? 6 cents ? 24 cents ? 

51. In 16 how many 2's? 8's? 4's? 

52. In 40 how many 5's ? 8's ? lO's ? 4's ? 

53. In 120 how many lO's? 12's? 20's? 40's? 

54. In 72 how many 6's ? 12's ? 8's ? 9's ? 

55. In 60 how many lO's ? 6's ? 20's ? 5's ? 12's ? 





2. 






1. 


How many 3's in 6 ? In 9 ? In 18 ? 


In 24? 


In 21? 


2. 


How many 4's in 8 ? In 20 ? In 32 ? 


In 48? 


In 20? 


3. 


How many 5's in 10 ? In 30 ? In 20 ? 


In 40? 


In 60? 


4. 


How many 6's in 1 8 ? In 42 ? In 54 ? 


In 72? 


In 24? 


5. 


How many times is 7 contained in 21 ? 








In 28? In 14? In 77? In 


35? 


In 56? 




In 42? In 63? In 70? In 


49? 


In 84? 


6. 


How many times is 8 contained in 24 ? 








In 40? In 56? In 72? In 


88? 


In 80? 




In 16? In 32? In 48? In 


64? 


In 96? 


7. 


How many times is 9 contained in 54 ? 








In 18? In 45? In 72? In 


99? 


In 27? 




In 81? In 108? In 36? In 


63? 


In 90? 



' In 110? In 30? 
? In 40? In 70? 


In 66? 
In 99? 


In 110? 
In 132? 


In 108? 
In 120? 


In 132? 
In 144? 



34 MENTAL ARITHMETIC. 

8. Howmany 10'sin60? 

In 20? In 50? In 80? 
In 100? In 90? In 120? 

9. How many ll's in 33? 

In 77? In 121? In 22? 
In 44? In 88? In 55? 

10. How many 12's in 24? 
In 36? In 60? In 84? 
In 48? In 72? In 96? 

11. How many 2's in 7 ? 

Solution. — In 7 there are three 2*8, and a remainder of 1. 

12. In 13 there are how many 2's? 3's? 4's? 5's? 

13. In 14 there are how many 2's? 4's? 6's? 7's? 

14. In 15 there are how many times 2? 3? 4? 5? 6? 

15. 16 is how many times 2? 3? 4? 5? 6? 8? 

16. 17 is how many times 3? 4? 6? 5? 7? 8? 

17. In 18 there are how many times 2? 4? 5? 6? 9? 

18. 19 is how many times 3? 5? 6? 8? 9? 

19. 20 is how many times 2? 3? 5? 6? 8? 10? 

20. 25 is how many times 2? 3? 6? 5? 8 ? 10? 

21. 30 is how many times 3 ? 5? 6? 10? 4? 8? 

22. 35 is how many times 4? 5? 6? 7? 8? 10? 

23. 40 is how many times 4? 5? 8? 10? 6? 7? 

24. 50 is how many times 5? 10? 12? 8? 

25. 60 is how many times 5 ? 6? 10? 12? 7? 

26. 70 is how many times 7? 10? 8? 9? 12? 

27. 80 is how many times 8? 10? 20? 4? 12? 

28. 90 is how many times 9 ? 10 ? 30? 18 ? 5? 

29. 100 is how many times 10? 4? 20? 30? 25? 

30. 100 is how many times 50? 2? 8? 12? 11? 

Note. — If further exercises are needed, the numbers omitted from 
20 to 100 may be used similarly to the foregoing. 



DIVISION. 35 



1. If 2 boys earn 6 dollars, how much does one boy 
earn? 

Solution. — If 2 boys earn 6 dollars, one boy earns as many dollars 
as 2 is contained times in 6, or 3 dollars. 

Note. — By some arithmeticians questions like the foregoing are 
called questions in partition. They are more readily solved by frac- 
tions, as follows: 

If 2 boys earn 6 dollars, one boy earns J of 6 dollars, or 3 dollars. 

2. If 5 boys have 30 marbles, how many have they 
apiece? 

3. John shares 8 apples among 4 boys ; how many does 
each receive? 

4. If 3 oranges cost 12 cents, how much does 1 orange 
cost? 

5. If 5 cords of wood cost 20 dollars, how much does 1 
cord cost? 

6. I paid 12 cents for 6 peaches ; what was the cost of 1 
peach ? 

7. If 10 cords of wood cost 50 dollars, how much will 1 
cord cost? 

8. A man paid 40 dollars for 8 sheep ; how much did they 
cost apiece ? 

9. A man drove 63 miles in 9 hours ; how far did he drive 
in 1 hour? 

10. George earned 21 dollars in 7 days ; how much did he 
earn in 1 day? 

11. How many oranges at 4 cents each can I get for 6 
lemons at 2 cents each? 

12. I exchange 12 pencils worth 5 cents apiece for slates at 
10 cents apiece ; how many slates do I get ? 

13. Henry exchanged 6 bananas worth 3 cents each for 
lemons at 2 cents each ; how many lemons did he get ? 



36 MENTAL ARITHMETIC. 

14. How many bananas at 5 cents each can I get for 3 
dozen eggs at 10 cents a dozen? 

15. I exchange 6 tons of coal at 5 dollars a ton for hay at 
10 dollars a load ; how many loads of hay do I get? 

16. A merchant sold 10 pounds of sugar at 9 cents a pound, 
and took in exchange 3 pounds of butter ; how much a pound 
was the butter worth ? 

17. How many pounds of rice at 8 cents a pound can I get 
for 4 dozen of eggs at 20 cents a dozen ? 

18. A drover exchanged 3 horses worth 120 dollars each 
for cows worth 40 dollars each ; how many cows did he 
get? 

19. A boy sold 30 quarts of chestnuts to a merchant at 8 
cents a quart, and took in exchange coffee at 20 cents a 
pound; how many pounds of coffee did he get? 

20. If James gives 7 quarts of nuts worth 8 cents a quart 
for 6 pounds of sugar at 9 cents a pound, how much does he 



21. A boy exchanged 5 dozen eggs at 18 cents a dozen 
for 4 dozen buttons at 20 cents a dozen ; how much cash 
ought he to receive? 

22. How many are 8 plus 4, divided by 3 ? 10 plus 8, 
divided by 3 ? 12 plus 12, divided by 6 ? 

23. How many are 10 plus 20, divided by 5? 16 plus 12, 
divided by 7? 15 plus 12, divided by 9 ? 

24. How many are 18 + 14, divided by 8? 20 + 20, di- 
vided by 5? 18 + 30, divided by 12? 25 + 19, divided 
by 11? 

25. How many are 18 minus 8, divided by 2? 27 minus 
11, divided by 8? 30 minus 6, divided by 6? 

26. How many are 72-12, divided by 12? 75-25, di- 
vided by 10? 49 — 5, divided by 11? 

Note. — The sign of division is -^^ ; it is read " divided by." Thus 
16 -f- 4 is read 16 divided by 4. 



DIVISION. 37 

27. How many are 60 - 20, -^ 5 ? 40 — 10, ^ 6 ? 24 - 2, 
-^11? 39-3,^12? 

28. How many are 6 times 5, divided by 3 ? 7 times 8, 
divided by 14? 6 times 15, divided by 30? 

29. How many are 7 X 12, divided by 21 ? 6 X 15, di- 
vided by 9? 8X12, divided by 16? 

30. How many are 10 X 8, -^ 20 ? 6 X 16, ^12 ? 3 X 24, 
-^9? 6X14,-^7? 2X35,^14? 

31. James had 26 cents and found 10 more; how many 
tablets at 9 cents each can he buy for his money ? 

32. John had 15 dollars and Henry had 20 ; how many 
sheep can they together buy at 7 dollars apiece ? 

33. If I earn 27 cents and my father give me 21 more, 
how many pencils can I buy at 4 cents apiece? 

34. Mary had 60 cents, but lost 6 ; how many pounds of 
sugar can she buy at 9 cents a pound ? 

35. A man having 60 dollars paid a debt of 24 dollars ; 
how many yards of cloth at 4 dollars a yard can he buy for 
the remainder of his money ? 

36. A man worked 6 weeks at 12 dollars a week, paying 5 
dollars a week for his board ; how many pairs of boots at 7 
dollars a pair can he buy with what he saved ? 

37. Twice Mary's age + 3 times her age, equals 50 years ; 
how old is she ? 

38. Four times my age + 2 times my age, — 3 times my 
age, is 36 years ; how old am I ? 

39. Three times a number, -f 5 times the number, — 2 
times the number, + 4 times the number, equals 90 ; what is 
the number? 

40. Five times a number, multiplied by 4, divided by 10, 
equals 24 ; what is the number ? 

41. How many are 20 + 8 — 16 -^ 4 ? 

Note. — In problems like this, containing the signs +> — > and X or 
-?-, the indicated multiplication or division should be performed before 



38 MENTAL ARITHMETIC. 

the additions or subtractions are made. Thus : 20 + 8 — 1 6 ^- 4 = 20 
+ 8 — 4 = 24. 

Problems like the following : 30 -J- 5 X 3, where the signs of multi- 
plication and division occur in succession, are ambiguous. Thus, 30 -r- 

30 30 

5X3 may mean — X 3 = 18, or it may mean = 2. 

5 5X3 

It is better in such questions to use a comma to convey the meaning, 

Thus, 30 H- 5, X 3 == 18, or 30, -T- 5 X 3 = 2. The vinculum also is used 

for this purpose, also curved lines ; as 30 -f- 5 X 3 = 18, or (30 -r- 5) X 

3 = 18. 

Find the value of the following: 

42. 4 + 8-3 + 6. 50. 24X3-4X12. 

43. 28-4 + 16-5. 51. 40-10X2 + 32. 

44. 25 + 7-14-8. 52. 16-^4 + 6-2. 

45. 30-16 + 2-10. 53. 28^4-6 + 5. 

46. 24 + 16-18+3. 54. 14+6^2-4. 
47.16 + 8X2. 55.27 + 3-16-^2. 
48.20 + 10X3-14. 56.3X4 + 6X9. 
49. 18 X 2 - 6 + 8. 57. 18 ^3 + 6 - 5. 

58. 63 -f- 7 + 6 X 4 - 27 -f- 9. 

59. 48^6 + 3X5-30-^3. 

60. 48x2-10 + 5-16^4. 

61. 120^5+5X6-18-^3+3X7. 

62. 9,X8, -^6, X3,-16. 

63. 45, -^ 5, X 8, + 3, -^ 15, + 8. 

64. 20, X 4, + 10, ^ 6, + 15, -^ 5, + 6. 

65. 24, - 4, + 16, ^ 3, + 6, -^ 9. 

66. (7 + 17 ) -^ 6 + (20+_16) -^- 9. 

67. 48 + 6 ^ 6 - 18"+30~-^ 8. 



68. 4T7 X 8 - 44 ^ 11 + 16 X 3 - 18 ^ 6. 

69. (7 + 3) X 4 + (16 X 3) -^ 8 - (27 -^ 3) X 4. 

70. (16 X 4 - 18 X 2) ^ (14 X 2 - 3 X 7). 

71. 16, + 4, - 2, X 3, -^ 6, X 8, - 2, + 11, -f- 9. 

72. 8, X 10, + 4, -^ 7, X 6, + 8, -^ 20, X 10, -^ 4, + 26, - 
4, + 18, -^ 5. 



UNITED STATES MONEY. 39 



UNITED STATES MONEY. 

United States Money is the legal currency of the United 
States. 

Business transactions are usually conducted in dollars and 
cents. 

The following is the table : 

10 mills (M.) equal 1 cent, ct. . 
10 cents " 1 dime, d. 

10 dimes " 1 dollar, $. 

10 dollars " 1 eagle, E. 

Dollars and cents are written with a point ( . ) placed be- 
tween ; thus, $3.24 is read 3 dollars and 24 cents. 

1. How many mills in 2 cents ? In 5 cents ? In 4 cents? 
In 8 cents ? 

2. How many cents in 20 mills? In 60 mills? In 70 
mills ? 

3. How many cents in 3 dimes? In 6 dimes? In 8 
dimes ? 

4. How many dimes in 40 cents? In 50 cents? In 90 
cents ? 

5. How many dimes in $2 ? In $8 ? In S6 ? 

6. How many cents in $3 ? In $5 ? In $7 ? 

7. How many cents in $2.25 ? In $6.50 ? In $8.43 ? 

8. How many dollars in 70 dimes ? In 80 dimes ? 

9. How many dollars in 624 cents ? In 729 cents ? In 
942 cents? In 465 cents? 

10. How many dollars in 3 eagles ? In 7 eagles ? 

11. If I buy a book for 90 cents and a slate for 20 cents, 
how much do I pay for both ? 

12. If some sugar cost me 80 cents and some rice 55 cents, 
how much do both cost ? 



40 MENTAL ARITHMETIC. 

13. If a goose cost 90 cents and a pair of chickens 75 
cents, how much do both cost ? 

14. A turkey cost $1.25 and some hens $1.20; how much 
did both cost? 

15. Bought a pair of boots for $7.50 and a hat for $3.25 ; 
what was the cost of both ? 

16. I bought a ton of coal for $5.25 and some lumber for 
$10.50 ; how much did I pay for both ? 

17. If a horse cost $150 and a cow $37.50, how much do 
both cost ? 

18. Bought a hen for 45 cents, a pair of ducks for 90 cents, 
and 2 turkeys for $2.75 ; what was the cost of all ? 

19. If I buy a book for 60 cents and sell it for 85 cents, 
how much do I gain? 

20. Albert receives $2.50 a day and Henry $2.25; how 
much more does Albert receive in a day than Henry? 

21. A merchant bought cloth at $1.70 a yard and sold it 
at $2.25 a yard ; how much did he gain a yard ? 

22. A dealer bought chairs at 75 cents apiece and sold 
them at 95 cents apiece; how much did he gain on them? 

23. Bought books at $1.25 apiece and sold them at $1.60 
apiece ; what was the gain on each ? 

24. A horse cost $122.50 and a cow $34; how much was 
gained if both were sold for $160 ? 

25. Bought a coat for $18, a hat for $5.50 and a pair of 
shoes for $4.25: if I paid for them with $30, how much 
change did I get? 

26. Henry had $40.25: he paid $16.50 for board and 
$14.25 for books ; how much had he remaining ? 

27. How much will 3 hens cost at 40 cents each ? 

28. If a boy can earn 60 cents a day, how much can he 
earn in 5 days ? How much in 8 days ? 

29. A drover sold sheep at $4 each ; how much did he re- 
ceive for 20 sheep ? 



UNITED STATES MONEY. 41 

30. How much are 25 sheep worth at $4.50 each ? 

31. What is the cost of 12 books at $1.25 each ? 

32. Bought 12 hens at 30 cents each and 4 ducks at 35 
cents each ; how much did they all cost ? 

33. A lady, having $5, bought 4 chickens at 40 cents each 
and a turkey for $1.50 ; how much money had she remaining? 

34. If I buy 20 books at $1.10 each and sell them at $1.25 
each, how much do I gain ? 

35. What is the cost of 7 pounds of coffee at 21 cents a 
pound and 4 pounds of sugar at 9 cents a pound ? 

36. A man bought 3 pounds of butter at 30 cents a pound, 
a bag of flour for 90 cents, and 5 pounds of coffee at 25 cents 
a pound : he paid for all with a 5-dollar bill ; how much 
change did he get? 

37. If 5 pounds of sugar cost 35 cents, how much does a 
pound cost ? 

38. I paid $1.20 for 3 hens ; how much was that apiece? 

39. If 40 sheep cost $160, what is the cost of 1 sheep? 

40. I paid $6 a dozen for books and sold them at 75 cents 
apiece ; how much did I gain on each ? 

41. If chairs that cost $15 a dozen are sold at $1.50 each, 
what is the gain ? 

42. Bought 30 sheep for $120 and sold them at $4.75 
apiece; what was the gain on each and on all? 

43. Samuel bought 25 hens for 6 dollars and sold them at 
a gain of 8 cents apiece ; how much did he get for them ? 

44. A farmer bought 12 acres of land for 1500 dollars and 
sold it for $1800 ; how much did it cost him an acre, and how 
much did he gain per acre ? 

45. If pencils cost 72 cents a dozen, how much is that 
apiece ? 



42 MENTAL ARITHMETIC. 



FRACTIONS. 

If anything, as an apple, is divided into two equal parts, 
one of these parts is called a half; if into three equal parts, 
each part is called a third ; if into four equal parts, each part 
is called a fourth, and so on. These equal parts of a unit, 
one-half (|), one-third (^), one-fourth (|), etc., are called 
fractions, 

1. How many halves in 1 orange ? In 2 oranges ? In 5 
oranges ? 

2. What is one-half of 8 books? 

Arudysis. — One half of 8 books is 4 books, because 2 times 4 books 
are 8 books. 

3. What is 1 half of 6 boys ? Of 4 cents ? Of 8 dollars ? 

4. What is 1 half of 10 roses ? Of 12 hens ? Of 14 toys ? 

5. What is 1 half of 20 men ? Of 1 6 ducks ? Of 30 pens ? 

6. What is 1 half of 6? Of 22? Of 24? Of 40? 

7. What is 1 half of 18 ? Of 32 ? Of 50 ? Of 60 ? 

8. If a pound of rice cost 8 cents, how much will 1 half 
pound cost? 

9. George has 18 marbles, and Henry has 1 half as many ; 
how many has Henry ? 

10. If a dozen eggs cost 20 cents, how much will 1 half 
dozen cost ? 



11. How many thirds in an apple? How many in 2 ap- 
ples ? In 5 apples ? 

12. What is I of 6 cents? Of 12 cents? Of 15 cents ? 

13. What is 4 of 9 men ? Of 27 men ? Of 36 men ? 

14. What is 1 of 21 ? Of 24? Of 30? Of 33? 

15. What is i of 12? Of 36? Of 48? Of 60? 



FRACTIONS. 43 

16. A book cost 60 cents and a slate 1 third as much ; how 
much did the slate cost ? 

17. Mr. Smith is 36 years old and his son is 1 third as old ; 
how old is the son ? 

18. James has 27 cents and John has 1 third as many; 
how many has John? 

19. William earned 54 dollars in a month and his brother 
1 third as much ; how much did the brother earn ? 

20. How many are 2 thirds of 6 apples ? 

Solution. — 1 third of 6 apples is 2 apples, and 2 thirds of 6 apples 
are 2 times 2 apples, or 4 apples. 

21. How many are 2 thirds of 9 men? Of 12 men ? Of 
18 men ? 

22. How many are 2 thirds of 15 dollars? Of 24 dollars? 
Of 30 dollars? 

23. How many are 2 thirds of 6 ? Of 15 ? Of 21 ? 

24. How many are 2 thirds of 12 ? Of 60 ? Of 45 ? Of 
48? 

25. Harry earned 30 dollars a month and Amos 2 thirds 
as much ; how much did Amos earn ? 

26. Bought 24 pencils and sold 2 thirds of them ; how 
many did I sell ? 

27. A man earned 45 dollars and spent 2 thirds of it ; how 
much did he spend ? 

28. If I can earn 48 dollars in a month, how much can I 
earn in 2 thirds of a month ? 

29. One book cost 66 cents and another 2 thirds as much ; 
how much did the second book cost ? 



30. How many fourths in 1 apple ? In 2 apples ? In 3 
apples ? In 6 apples ? 

31. What is 1 fourth of 4 cents? Of 8 cents? Of 20 
cents ? 



44 MENTAL ARITHMETIC. 

32. What is 1 fourth of 12 ? Of 16 ? Of 24 ? Of 48 ? 

33. What is 1 fourth of 32 ? Of 20 ? Of 36 ? Of 60 ? 

34. A cow cost 24 dollars and a sheep 1 fourth as much ; 
how much did the sheep cost? 

35. If a man earn 32 dollars and spend 1 fourth of it, how 
much does he spend ? 

36. A farmer had 40 acres of land, 1 fourth of which he 
planted in corn ; how many acres did he plant in corn ? 

37. How much is 2 fourths of 16 dollars? Of 12 cents? 
Of 20 men? 

Note. — Fractions are generally written as figures alone; thus 2 
fourths is J, 3 fifths is f , 5 sevenths is f, and so on. 

38. How much is | of 8? Of 24? Of 32? Of 44? 

39. A man having 28 bushels of corn sold f of it ; how 
much did he sell ? 

40. Harry having 40 cents spent | of it ; how much did 
he spend ? 

41. How much is | of 20 cents? Of 48 apples? Of 36 
men? 

42. How much is | of 12? Of 16? Of 40? Of 28? 
Of 32? 

43. How much is | of 48? Of 60? Of 80? Of 36? 
Of 72? 

44. George earns 48 dollars a month and Henry earns | as 
much ; how much does Henry earn ? 

45. A man had $60 and gave | of it for a cow ; how much 
did he pay for the cow ? 

46. Henry is 20 years old and his brother is | as old ; how 
old is his brother ? 

47. If a man pay | of $1.20 for a book, what does the 
book cost him? 

48. James had $36, William had | as much, and Andrew 
I as much ; how much had William and Andrew each ? 



FRACTIONS. 45 

1. If an apple is divided into 5 equal parts, what is 1 of 
these parts called ? 2 parts ? 3 parts ? 

2. How many fifths in a pie ? In 3 pies ? In 5 pies ? 

8. What is 1 fifth of 10 cents? Of 20 cents? Of 35 
cents ? 

4. What are | of 10 ? Of 20 ? Of 80 ? Of 40 ? 

5. What are f of 15 ? Of 30 ? Of 50 ? Of 45 ? 

6. What are | of 20? Of 40? Of 55? Of 75? 

7. John earned 25 cents and spent f of it ; how much did 
he spend ? 

8. A knife cost 1 dollar and a book § as much ; what was 
the cost of the book ? 

9. James had $45 and earned f as much more ; how much 
had he then ? 

10. A pair of boots cost $10 and a pair of shoes ^ as much ; 
how much did the shoes cost ? 

11. A boy had 75 cents, but spent | of it ; how much did 
he spend ? 

12. If anything is divided into 6 equal parts, what is 1 of 
these parts called ? 3 parts ? 5 parts ? 

13. How many sixths are there in an orange? In 2 
oranges? In 5 oranges? In 10 oranges? 

14. How many are } of 12 boys? Of 30 cents? Of 42 
apples ? 

15. What are | of 18 ? Of 36 ? Of 48 ? Of 60 ? 

16. What are | of 24 ? Of 42 ? Of 54 ? Of 30 ? 

17. AVhat are f of 12 ? Of 30 ? Of 48 ? Of 72 ? 

18. What are | of 24 ? Of 66 ? Of 42 ? Of 54 ? 

19. Mr. Morgan is 48 years old and his son is | as old ; 
how old is the son ? 

20. If one man earn $60 and another earn f as much, how 
much will the second earn ? 

21. A boy, having 48 cents, spent | of it; how much did 
he spend ? 



46 MENTAL ARITHMETIC. 

22. If Mr. A has 36 acres of grass and | as many acres 
of corn, how many acres of corn has he ? 

23. John had 42 marbles and won | as many more ; how 
many had he then ? 

24. A cow cost $60 and a colt | as much ; how much did 
they together cost ? 

25. If an orange is divided into 7 equal parts, what is 1 of 
these parts called ? 2 parts ? 5 parts ? 

26. How many sevenths are there in a melon? In 2 
melons? In 3 melons? In 5 melons? 

27. What is } of 21 cents ? Of 35 dollars ? Of 28 cents ? 
Of 56 boys? 

28. What are § of 14? Of 70? 

29. What are I of 21? Of 35? 

30. What are ^ of 14 ? Of 28 ? 

31. What are ^ of 70? Of 84? 

32. Whataref of 49? Of 63? 

33. A cow was bought for $42 and sold for ^ of the cost ; 
at what price was she sold ? 

34. If I buy a watch for $70 and sell it at f of the cost, 
how much do I lose? 

35. Mr. A paid $84 for a carriage and ^ as much for a 
sleigh ; how much did both cost ? 

36. If a horse cost $140 and a sleigh f as much, how 
much is the loss if both are sold for $190 ? 

37. If a man earns $77 and pays f of it for board, how 
much has he remaining? 

38. 3 sevenths of $56 is $10 less than the cost of a cow ; 
how much did the cow cost ? 

39. f of $70 is $30 less than the cost of a sleigh ; how 
much did the sleigh cost? 

40. A man had $84 ; he spent 4 of it and gave | of the 
remainder to his sister ; how much did he spend, and how 
much did he give to his sister? 



Of 49? 


Of 35? 


Of 56? 


Of 63? 


Of 42? 


Of 56? 


Of 21? 


Of 35? 


Of 77? 


Of 28? 



FRACTIONS. 47 

1. If anything is divided into 8 equal parts, what is one 
of these parts called ? 3 parts ? 5 parts ? 7 parts ? 

2. How many eighths in a pie ? In 2 pies ? In 4 pies ? 

3. What is | of 16 roses? Of 32 pinks? Of 40 
flowers ? 

4. What are | of 16 ? Of 32 ? Of 48 ? Of 64 ? 

5. What are | of 8? Of 24? Of 56? Of 72? 

6. What are | of 24? Of 40? Of 56? Of 80? 

7. What are I of 32? Of 48? Of 64? Of 96? 

8. If a carriage cost $80 and a set of harness | as much, 
how much does the harness cost ? 

9. A young lady wrote 72 letters one day and | as many 
the next ; how many did she write the second day ? 

10. If James has 96 cents and George has f as many, how 
many have they together ? 

11. Samuel has 64 cents and William has | as many ; how 
many has Samuel more than William? 

12. If anything is divided into 9 equal parts, what is one 
of these parts called? 3 parts? 7 parts? 

13. How many ninths of an apple in 1 apple? In 2 
apples? In 5 apples? 

14. How much is ^ of 36 men? Of 45 pupils? Of 63 
horses? Of 81 questions? 

15. What are § of 18 ? Of 36 ? Of 54 ? Of 72 ? 

16. What are | of 27 ? Of 45 ? Of 9 ? Of 63 ? 

17. What are f of 72 ? Of 81 ? Of 90 ? Of 99 ? 

18. What are J of 18 ? Of 27 ? Of 45 ? Of 108 ? 

19. What are | of 36 ? Of 63 ? Of 90 ? Of 81 ? 

20. A cow cost 45 dollars and a colt § as much ; how much 
did the colt cost ? 

21. William had 36 marbles and Henry gave him | as 
many more ; how many had he then ? 

22. Alfred had 90 marbles and gave | of them away ; how 
many had he remaining ? 



Of 60 cents? Of 40 dol- 


Of 50? 


Of 70? 


Of 90? 


Of 80? 


Of 120? 


Of 60? 


Of 100? 


Of 70? 


OfllO? 



48 MENTAL ARITHMETIC. 

23. James had 63 cents ; he earned | as much more, and 
then spent 42 cents; how much money had he remaining? 

24. Harry had § of $36 and his father had 5 times as 
much ; how many dollars had the father ? 

25. When anything is divided in 10 equal parts, what is 1 
part called ? 4 parts ? 6 parts ? 

26. What is ^V of 40 cents? 
lars? Of 100 peaches? 

27. How many are ^q of 30 ? 

28. How many are -f^ of 40? 

29. How many are j% of 20 ? 

30. I had 70 cents, but I gave -^q of it for a slate ; how 
much money have I remaining? 

31. A drover had 120 sheep ; he sold ^ of them to one 
man and y\ to another ; how many did he sell to each ? 

32. How much will an arithmetic cost if a geography cost 
$1.20 and an arithmetic y\ as much? 

33. John, having 80 cents, earned /^ as much more, and 
then spent j\ of all his money ; how much did he spend ? 

34. I of 24 are how many times 4 ? 

35. I of 40 are how many times 6 ? 

36. I of 42 are how many times 5 ? 

37. ^ of 63 are how many times 6 ? 

38. f of 63 are how many times 5 ? 

39. f of 72 are how many times 3 ? 

40. I of 80 are how many times 25 ? 

41. ^ of 81 are how many times 15? 

42. 5 of 72 are how many times 4 ? 

43. -j^ of 60 are how many times 3 ? 

44. 5 of 90 are how many times i of 50 ? 

45. /^ of 80 are how many times ^ of 18 ? 

46. -^ of 90 are how many times J of 35 ? 

47. I of 96 are how many times | of 25 ? 

48. I of 54 are how many times f of 30 ? 



FRACTIONS. 49 



49. f of 84 are how many times | of 24 ? 

50. I of 63 are how many times ^ of 49 ? 



2. 

KEDUCTION OF FRACTIONS. 

1. How many halves in an apple? How many thirds? 
How many fourths? How many fifths? 

2. How many halves in 3 oranges ? 

Solution. — In 1 orange there are 2 halves, and in 3 oranges there 
ate 3 times 2 halves, or 6 halves. 

3. How many halves in 4 ? In 6 ? In 5 ? In 8 ? 

4. How many thirds in 3 ? In 7 ? In 4 ? In 10 ? 
5 How many fourths in 2 ? 3 ? 5 ? 4 ? 6 ? 

6. How many fifths in 3? 5? 6? 7? 9? 

7. How many sixths in 4 ? 5 ? 8 ? 10 ? 4 ? 

8. How many sevenths in 5? 3? 9? 7? 11? 

9. How many eighths in 4 ? 6 ? 8 ? 10 ? 12 ? 

10. How many ninths in 3 ? 5 ? 7 ? 9 ? 11 ? 

11. How many tenths in 4? 5? 8? 10? 12? 

12. How many thirds in 5| ? 

Solution. — In 1 there are 3 thirds ; in 5 there are 5 times 3 thirds, 
or 15 thirds ; 15 thirds and 1 third are 16 thirds. Therefore in 5 J there 
are 16 thirds. 

13. How many fourths in 3 ? In 5 ? In 2| ? 

14. How many fifths in 2 ? In 6 ? In 3| ? In 4^ ? 

15. How many sixths in 3 ? In 2i ? In 3| ? 

16. How many sevenths in 4? In 3f ? In 5^? 

17. How many eighths in 3? In 4^ ? In 5| ? 

18. How many ninths in 4 ? In 3^ ? In 7| ? 

19. How many tenths in 5 ? In Qj\ ? In 8jV? 

20. Reduce 4| to an improper fraction. 

• 4 



50 MENTAL ARITHMETIC. 

Note. — An improper fraction is one whose denominator is less than 
its numerator. 

Solution. — In 1 there are 5 fifths ; in 4 there are 4 times 5 fifths, or 
20 fifths ; 20 fifths plus 3 fifths are 23 fifths. 

21. Keduce to improper fractions 2| ; 3J ; 4| ; 4|. 

22. Reduce to improper fractions 3^ ; 5^; 6| ; 8^. 

23. Reduce to improper fractions 4^; 5| ; 6| ; 7|. 

24. Reduce to improper fractions 3^ ; 4| ; 6f ; 7|. 

25. Reduce to improper fractions 6^ ; 7^^ ; 5| ; 8f . 

26. If 2^ yards of cloth cost 14 dollars, how much will a 
yard cost? 

Solution. — 2J yards are equal to J yards. If J yards cost 14 dol- 
lars, ^ of a yard will cost } of 14 dollars, or 2 dollars ; and |, or a 
yard, will cost 3 times 2 dollars, which are 6 dollars. 

27. If 6A pounds of sugar cost 52 cents, how much does 1 
pound cost? 

28. How much will a bushel of beans cost if 2| bushels 
cost $8.10? 

29. I paid 55 cents for 3| dozen of eggs ; how much was 
that for 1 dozen ? 

30. If 5J tons of coal cost 33 dollars, how much will 1 ton 
cost? 

31. I paid 34 cents for 2| yards of gingham ; what was the 
price per yard? 

32. A man can earn 40 cents in 2§ hours ; how much can 
he earn in 10 hours ? 

33. A train of cars runs 60 miles in 2^ hours ; how far 
does it go in 8 hours? 

34. If I pay 44 cents for 2| pounds of meat, how much 
ought I to pay for 10 pounds ? 

35. George earns 100 dollars in 3^ months ; how much can 
he earn in 12 months? 

36. There are 132 minutes in 2i hours ; how many minutes 
in 6 hours ? 



FRACTIONS. 61 

37. In 3| days there are 78 hours ; how many hours in 2 
days? 

38. If 2| pounds of cheese cost 38 cents, what are 5 pounds 
worth ? 

39. What is the cost of 3 pounds of butter if 2| pounds 
cost 63 cents ? 

40. A boy can walk 16 miles in 5^ hours ; how far can he 
walk in 12 hours? 



3. 

1. How many ones in | ? 

Solution. — Since 1 equals f , in f there are as many ones as f is 
contained times in |, or 2. 

2. How many ones in | ? In -i/- ? In -\^- ? In J/ ? 

3. How many ones in | ? In J^ ? In ^O-? In ^^^- ? 

4. How many ones in -i^? In -2^? In ^? In J/-? 

5. How many ones in -\^? Y? i^*^ U^ 

6. Reduce -V^- to a mixed number. 

Note. — A mixed number consists of a whole number and a fraction. 
Solution. — Since 1 equals f , in J/ there are as many ones as f is 
contained times in ^^-, or 2f . 

7. Reduce to mixed numbers J^^- ; | ; -^5^ ; \^-, 

8. Reduce to mixed numbers J/ ; \'^- ; ^- ; ^^. 

9. Reduce to. mixed numbers -^; -V-; -^-', V~* 

10. Reduce to mixed numbers \^- ; -^3"^ ; -5^- ; -^^-. 

11. Reduce to mixed numbers -y-; ^^-; -^-; -j^-. 

12. Reduce to mixed numbers %^-; ^ ; -V- ; ^• 

13. How much will 5 pairs of shoes cost at 3| dollars a 
pair ? 

• Solution. — If 1 pair of shoes cost 3f dollars, 5 pairs will cost 5 
times 3f dollars. 5 times 3 dollars are 15 dollars ; 5 times f of a dol- 
lar are -^^ dollars, or 3f dollars ; 15 dollars plus 3f dollars are 18f dol- 
lars. Therefore 5 pairs of shoes at 3f dollars a pair will cost 18f 
dollars. 



62 MENTAL ARITHMETIC. 

14. How much will 9 barrels of apples cost at 2| dollars a 
barrel ? 

15. What is the cost of 6 pairs of shoes at 3| dollars a 
pair? 

16. If 4 dozen eggs cost 50 cents, how much does 1 dozen 
cost? 

17. If a boy can walk 16| miles in 5 hours, how far can 
he walk in 1 hour? 

18. What is the cost of 6 tons of coal at 5^ dollars a 
ton? 

19. I paid ly^^ dollars for 5 pounds of butter ; how much 
is that a pound ? 

20. If 3 hats cost 6| dollars, how much will 5 hats 
cost? 

21. How much can a man earn in 6 days at the rate of 2| 
dollars a day ? 

22. How much will 16 bushels of potatoes cost at the rate 
of I of a dollar a bushel ? 

23. If 4 pairs of shoes cost 164 dollars, how much is that 
a pair ? 

24. If a boy can earn 6| dollars in 5 days, how much can 
he earn in 3 days ? 

25. How far can a man walk in 10 hours, if he can walk 
5 miles in 2 hours ? 

26. At the rate of 5| dollars a ton, how much will 6 tons 
of coal cost? 

27. How much will 6 pounds of butter cost at 12^ cents a 
pound ? 

28. What is the cost of 8 pounds of rice at 6| cents a 
pound ? 



FRACTIONS. 53 



FRACTIONS TO HIGHER TERMS. 

1. How many sixths in ^ ? 

Solution. — In 1 there are f ; hence in J there are J of f , or f . 

2. How many fourths in ^ ? 

3. How many sixths in ^ ? ^ ? | ? 

4. How many eighths in ^ ? ^ ? | ? 

5. How many tenths in ^ ? i ? | ? f ? 

6. How many twelfths in^?'|?'|? |? |? 

7. How many fifteenths in J ? | ? -^ ? | ? 

8. How many sixteenths in ^ ? | ? | ? | ? 

9. How many eighteenths in | ? | ? | ? | ? 

10. How many twentieths in | ? | ? f ? y^^j ? 

11. How many twenty-fourths in f ? j\? y^^ ? | ? 

12. Reduce | to twelfths. 

Analysis. — J equals j% ; hence f equals 3 times y\, or y^^. 

13. Reduce ^, |, | to twelfths. 

14. Reduce f , |, | to fifteenths. 

15. Reduce |, |, | to sixteenths. 

16. Reduce |, j\, /^ to twentieths. 

1 7. Reduce f , |, -f% to thirtieths. 

18. Reduce ^, |, | to twenty-fourths. 

19. Reduce | to fifteenths. 

Since | = ^, by what number do we multiply both nume- 
rator and denominator of -§- to obtain /^? 

Note. — To reduce a fraction to higher terms, multiply both numera- 
tor and denominator of the fraction by that number which will produce 
the required denominator. 

20. By what must both numerator and denominator of | 
be multiplied to reduce it to 6ths? To 12ths? To 15ths? 

21. By what must both numerator and denominator of | 
be multiplied to reduce it to 8ths? To 12ths? To 20ths? 



54 MENTAL ARITHMETIC. 

22. Keduce A to sixths ; to eighths ; to tenths. 

23. Reduce f to lOths ; to 20ths ; to 30ths. 

24. Reduce | to 12ths; to 20ths; to 36ths. 

25. Reduce J and ^ to 12ths. 

26. Reduce |, |, and f to 40ths. 

27. Reduce ^, ^, | to 24ths. 

Note. — When fractions have the same denominator they are said to 
have a common denominator. 

28. Reduce ^ and | to a common denominator. 
Analysis. — A common denominator of halves and fifths is tenths; 

I equals y^^, and § equals ^. 

29. Reduce ^ and ^ to a common denominator. 

30. Reduce | and ^ to a common denominator. 

31. Reduce f and | to a common denominator. 

32. Reduce ^ and ^ to a common denominator. 

33. Reduce -| and | to a common denominator. 

34. Reduce | and ^ to a common denominator. 

35. Reduce | and f to a common denominator. 

36. Reduce | and -^^ to a common denominator. 

37. Reduce -^ and f to a common denominator. 

38. Reduce ^, ^, and ^ to a common denominator. 

39. Reduce ^, J, and ^ to a common denominator. 

40. Reduce ^, |, and ^ to sl common denominator. 

41. Reduce |, -J-, and ^^ to a common denominator. 

42. Reduce ^, |, and -^^ to a common denominator. 

43. Reduce ^, f , and -| to a common denominator. 

44. Reduce ^, |, and ^ to a common denominator. 



5. 

FRACTIONS TO LOWER TERMS. 
1. How many fourths in y^? 
Analysis. — Since J equals j\, there are as many 4ths in ^^ as ^^ is 
contained times in j^j, or |. 



FRACTFONS. 55 

2. How many halves in f ; | ; -^- ; f ; -4^- ? 

3. How many thirds in f ; f ; | ; t| ; ff? 

4. How many fourths in | ; | ; -fi^ ', 1% ; if ? 

5. How many fifths in y% ; ^% ; ■^% ; t% ; if ?  

6. How many sixths in y^ ; y^^ 5 T2 J A 5 f g ^ 

7. How many sevenths in j\ ; 1^4 ; if > irr » 2t ^ 

8. How many eighths in f-^; /V; jg ; f^; 3z\? 

9. How many ninths in y^g ; y^g ; ^| ; || ; ^f ? 

10. How many tenths in .f^ ; 3%; ^% ; if ; ^g ? 

Since y^^r equals |, by what do we divide both numerator 
and denominator of y®^ to produce | ? 

Note. — To reduce a fraction to lower terms, divide both numerator 
and denominator by a number that will divide each. 

11. By what may we divide both numerator and denomi- 
nator of |? -p%? -y-? 

12. Reduce | to thirds; jf to 5ths. 

13. Reduce | to 3ds ; -^^ to 4ths. 

14. Reduce j% to 6ths ; -\^- to 4ths. 

15. Reduce II to 9ths; || to 6ths. 

16. Reduce y®g to 4ths; Jf to 5ths. 

17. Reduce ^f to 7ths ; J-^ to 3ds. 

18. Reduce y to 4ths; i| to lOths; /o to 5ths. 

Note. — When a fraction cannot be reduced to one having a less de- 
nominator, it is said to be in its lowest terms. 

19. Reduce j§ and | to their lowest terms. 

20. Reduce f^ and o^ to their lowest terms. 

21. Reduce y% and y| to their lowest terms. 

22. Reduce 3^^^ and ^J to their lowest terms. 

23. Reduce |-| and y^^ to their lowest terms. 

24. Reduce |g and |§ to their lowest terms. 

25. Reduce j| and ^f to their lowest terms. 

26. Reduce f f and f § to their lowest terms. 

27. Reduce J§ and ^^j to their lowest terms. 

28. Reduce ^f and |f to their lowest terms. 



56 MENTAL ARITHMETIC. 

6. 

COMPOUND FRACTIONS. 
A compound fraction is a fraction of a fraction ; as, ^ of | ; 

5 



|of ^- 



1. What is iof I? 

Analysis. — Since | equals f, J of | is J of |, or J. 

2. Wliatis^ of ^? -J of I? 

3. What is J of ^ ? | of |? i of | ? 

4. What is ^ of I ? ^ of ^ ? i of ^ ? 

5. What is i of J? i of ^? iof i? 

6. Susan had ^ a dollar and gave ^ of it away ; how much 
did she give away ? 

7. If I had I of a dollar and spent J of it, how much did 
I spend ? 

8. A man sells 5 of ^ of his corn ; how much does he sell ? 

9. How much is ^ of ;| of an inch? 

10. A man earned ^ of a dollar and his son ^ as much ; 
how much did the son earn ? 

11. How much is I of f ? 

Analysis. — J of } is yV ; J of | is 2 times as much, or ^^ ; and | of 
f is 2 times A, or j%. 

12. What is I of i ? 16. What is ^ of §? 

13. What is I of f? 17. What is | of | ? 

14. What is I of I ? 18. What is | of § ? 

15. What is I of I ? 19. What is | of | ? ' 

20. A boy had | of a dollar and spent | of it ; how much 
did he spend ? 

21. If a boy has 5 of a pie and gives away | of it, how 
much does he give away ? 

22. If Amos has f of a dollar and spends | of it, how much 
does he spend ? 



FRACTIONS. 57 

23. If I give away | of | of my money, how much do I 
give away ? 

24. A boy had 40 cents : he spent | of y% of it ; how much 
did he spend ? 

25. A horse cost 120 dollars, and a cow cost | of | as 
much ; how much did the cow cost ? 

26. A gentleman who owned | of a farm sold | of his 
share ; how much of the farm did he sell ? 

27. George had | of a bushel of chestnuts: he shared 
them equally among 6 of his playmates ; what part of a 
bushel did each get? 

28. AVilliam had -| of a dollar, and shared it with 3 broth- 
ers ; what part of a dollar did each get ? 



ADDITION OF FRACTIONS. 

1. What is the sum of | of a pie and | of a pie? 

2. What is the sum of | of a dollar and f of a dollar ? 

3. What is the sum of f of a dollar and f of a dollar? 

4. What is the sum of f of an orange and | of an orange ? 

5. How much is I of a peck, | of a peck, and f of a 
peck? 

6. What is the sum of I, |, and | ? 

7. What is the sum of f, |, and f ? 

8. What is the sum of ^, |, and f ? 

9. What is the sum of ^V, j\y and -^2 

10. How many 4ths in ^ and \ ? 

11. How many 6ths in ^ and | ? 

12. How many 8ths in ^ and | ? 

13. How many lOths in ^ and | ? 

14. How many 12ths in |, |, and J? 

15. How many 15ths in | and f ? 

16. How many 20ths in |, |, and ^^q ? 



58 MENTAL ARITHMETIC. 

17. What is the sum of ^ aod -|? 

Solution.— f equals |J, and | equals y% ; {^ plus -f^ are |^f, or 1^. 
What is the sum — 



18. 


Of 1 and 1 ? 


29. 


Of 1 and 1? 


19. 


Of ^ and 1 ? 


30. 


Of 1 and 1? 


20. 


Of i and i? 


31. 


Of i and § ? 


21. 


Of 4 and^? 


32. 


Of f and^? 


22. 


Of J and ^ ? 


33. 


Of 1 and ^? 


23. 


Of 1 and ^ ? 


34. 


Of iiandl? 


24. 


Of 1 and 1 ? 


35. 


Of ^.i.andi? 


25. 


Offandi? 


36. 


Of |,f and^? 


26. 


Of 1 and 1? 


37. 


Of i.^andf? 


27. 


Of 1 and J ? 


38. 


Of il.andi? 


28. 


Of f and J ? 


39. 


Of f , I and 1 ? 



40. What is the sum of 2| and 1| ? 

Solution 1. — 2f equals f ; IJ equals f ; f equals }f, and { equals 
11; fl plus 1^ equals fj, or 3}^. 

Solution 2. — 2 plus 1 are 3 ; J equals j% and | equals j^ ; /j plus 
y\ are {^ ; hence 2f plus IJ equals 3}|. 

Note. — In solution 2 the whole numbers and the fractions are added 
separately. 

What is the sum — 

41. Of 2i and 3i ? 46. Of 6^ and 5J ? 

42. Of If and 3|? 47. Of 4^ and 2|? 

43. Of 3| and 2|? 48. Of 8| and 3|? 

44. Of 2i and 3^? 49. Of 10| and 61? 

45. Of 6i and 2| ? 50. Of 6^ and 10|? 

51. I paid I of a dollar for a hen and f of a dollar for a 
duck ; how much did I pay for both? 

52. If a book cost f of a dollar and a slate | of a dollar, 
how much do both cost? 

53. If I pay ^ of a dollar for a dinner and | of a dollar 
for car-fare, how much do I pay for both ? 



FRACTIONS. 59 

54. George gave | of his money for oranges and ^ for 
candy ; how much of his money did he spend ? 

55. If a book cost | of a dollar, some paper ^^ of a dollar, 
and some pencils 3% of a dollar, how much do all cost ? 

56. Henry, having | of a dollar, earned | of a dollar ; how 
much had he then ? 

57. If l of a number increased by I of the same number 
equals 36, what is the number ? 

58. My money increased by -| of my money is 64 cents ; 
how much money have I? 

59. I of John's marbles increased by | of his marbles 
equals 39; how many marbles has he? 

60. Mary's number of pinks increased by | of her num- 
ber equals 45; how many pinks has she? 

61. Albert, having | of a certain sum of money, found -| 
of the same sum, and then had 81 cents ; what was the sum ? 

62. f of a certain number increased by f of the number 
is 74 ; what is the number ? 

63. George paid 3- of a dollar for a slate, | of a dollar for 
a book, and ^\ of a dollar for paper ; how much did he pay 
for all? 

64. If a man pays $3| for a pair of shoes and $U for a 
pair of gloves, how much does he pay for both ? 

65. If a man walk 3| miles the first hour and 3^ miles the 
second, how far does he walk in the two hours ? 

66. Bought a hat for $^, a pair of shoes for $5-J, and an 
umbrella for $2| ; how much did I pay for all ? 

67. Bought coal for $5| and wood for $^ ; how much did 
I pay for both ? 

68. I paid f of a dollar for ribbon, | of a dollar for mus- 
lin, and $2 J for gloves ; how much did I pay for all ? 

69. A boy walked 12y^o miles in the forenoon and 15^ 
miles in the afternoon ; how far did he walk during the 
day? 



60 MENTAL ARITHMETIC. 

70. James has $5 J, George has $6|, and Henry $3f ; how 
much money have they together? 



SUBTEACTION OF FRACTIONS. 

1. What is the difference between | of a dollar and | of a 
dollar? 

2. What is the difference between | of an apple and § of 
an apple? 

3. What is the difference between J of an acre and f of 
an acre? 

4. What is the difference between j*^^ of a cent and -^\ of 
a cent ? 

5. What is the difference between /^ of a dozen and j'^ of 
a dozen ? 

What is the difference — 
6 



Between y\ and j\ ? 


9. 


Between ^^ and -^^ ? 


Between y\ and fy ? 


10. 


Between y®^ and -^j ? 


Between if and y^g ? 


11. 


Between ^^ and ^§? 



12. How much is | — f ? 

Solution. — | equals -}^, and f equals y\ ; } J less y"^ is y\ ; hence 
f-f isxV 

What is the value — 

13. Of ^-|? 21. Of |-^? 

14. Of |-^? 22. Of 4-1? 

15. Of i- I? 23. Of ^-J? 

16. Of |-f? 24. Of ^-i? 

17. Of f- I? 25. Of |-f? 

18. Of |-|? 26. Of i--f? 

19. Of 1-1? 27. Of I -I? 

20. Of ^-}? 28. Of |-i? 
29. What is the difference between 2^ and If ? 
Solution. — 2f equals f ; If equals J ; f equals ff, and { equals j^l; 

ii — fi equals ^ ; hence the difference between 2f and If is yj. 



FRACTIONS. 61 

What is the difference — 

30. Between 2^ and If ? 34. Between 4| and 3f ? 

31. Between 3^ and 2|? 35. Between 7-^ and 6| ? 

32. Between 4^ and 3| ? 36. Between 3| and 2| ? 

33. Between 6^ and 3^ ? 37. Between 4i and 3^0 ? 

38. If a boy has | of a dollar and gives away | of a dollar, 
how much money has he remaining ? 

39. The difference between | of my money and f of it is 6 
dollars ; how much money have I ? 

40. Henry had $5^ and spent $3| ; how much money had 
he remaining? 

41. George had $5 and spent $21 ; how much had he re- 
maining ? 

42. If I buy sheep at $3| apiece and sell them at $4| 
apiece, how much do I make? 

43. Bought oranges at 2| cents each and sold them at the 
rate of 3| cents ; how much do I make on each ? 

44. James earned $6^ a week and William $5| dollars a 
week ; how much does James earn more than William ? 

45. A is 6 1 years old and B is 8^ years old; how much 
older is B than A ? 

46. Mr. Smith had 60 sheep : he sold | of them ; how many 
remained ? 

47. Mr. Martin is 40 years old and his son is | of | as old ; 
how old is his son ? 

48. Harry has 12 cents, and his number is f of | of Philip's 
number ; how many has Philip ? 

What is the value — 

49. Of 1+1-^? 53. Of l+f-i? 

50. Of ^+1-1? 54. Of i+j%-i? 

51. Of l+i-i? 55. Of l + l-^A,? 

52. Of I + ^ -I ? 56. Of 2^ + 3i -4| ? 

57. A man sold ^ of his sheep to one farmer, ^ to another, 
and had 6 remaining ; how many had he at first ? 



62 MENTAL ARITHMETIC. 

58. If I pay | of a dollar for one book and | of a dollar 
for another, and pay for both with a two-dollar bill, how much 
change do I get ? 

59. A hat cost $4J and a pair of shoes $3| ; how much 
change do I get in paying for them with 10 dollars? 

60. A merchant had 100 bushels of wheat in a bin : he 
sold to one man 24| bushels and to another man 13| bushels; 
how many bushels remain ? 



MULTIPLICATION OF FRACTIONa 

1. 

Fractions Multiplied by Integers. 

1. How much will 6 books cost at | of a dollar each? 

Solution. — Since 1 book costs | of a dollar, 6 books will cost 6 
times I of a dollar, or -^^ dollars, equal to 4^ dollars. 

2. At § of a dollar each, how much will 10 hens cost? 

3. George can earn f of a dollar in a day ; how much can 
he earn in 8 days ? 

4. William eats | of a dozen oranges in a day ; how many 
will he eat in 5 days ? 

5. If 1 dollar buy |^ of a yard of cloth, how many yards 
will $10 buy? 

6. How much will 12 bananas cost at the rate of f of a 
dime each ? 

7. How many are 6 times f ? 
Solution. — 6 times f are -V-, or 2f 
How many are — 

8. 3 times I? 12. 2 times f? 

9. 5 times I? 13. 3 times |? 

10. 4 times |? 14. 5 times f? 

11. 4 times |? 15. 6 times j% ? 



FRACTIONS. 63 

16. 4 times |? 18. 8 times y^? 

17. 6 times f? 19. 10 times |-? 

20. How mauy are 3 times f ? 

Solution. — 3 times J equals |, or |, and 3 times f equals 5 times 
i or f . 

Note. — Since 3 times f equals -\^-, or |, a fraction may be multiplied 
either by multiplying its numerator or by dividing its denominator. 

How many are — 

21. 3 times | ? 27. 7 times j\ ? 

22. 5 times y'^? 28. 8 times -j^^? 

23. 4 times f ? 29. 9 times /^^ ? 

24. 5 times |§ ? 30. 7 times /y ? 

25. 6 times j\ ? 31. 4 times 2| ? 

26. 6 times /^? 32. 3 times 6|? 

33. How much will 16 books cost at -f of a dollar each ? 

34. What is the cost of 12 bushels of oats at f of a dollar 
a bushel ? 

35. If a cow eat f of a bushel of bran in a day, how much 
will she eat in 4 days ? 

36. George earns y% of a dollar in one day ; how much 
will he earn in 12 days? 

37. At I of a dollar a pair, how much will 12 chickens 
cost? 

38. What is the cost of 4 tons of coal at $5i a ton ? 

39. I hired 8 men to work at $2| a day ; how much did I 
have to pay daily to all ? 

40. What is the cost of 8 pounds of sugar at f of a dime 
a pound? 

41. How far can a boy walk in 10 hours, if he can walk 
3| miles an hour? 

42. How much will 5 yards of silk cost at $2^ a yard ? 

43. What is the cost of 3 yards of cloth at $2| a yard? 

44. 5 times 5| years is twice my age ; how old am I ? 



64 MENTAL ARITHMETIC. 

45. 6 times ^5^ is 4 dollars less than a cow cost; how much 
did she cost? 

46. 5 times 3yV dollars is 3 dollars more than the cost of my 
coat ; how much did it cost ? 

47. 3 times 5J miles is ^ the distance from my home to the 
city ; how far is it to the city ? 



2. 

Integers Multiplied by Fractions. 

1. How much will ^ of a pound of nuts cost at 16 cents a 
pound ? 

Solution.— If 1 pound of nuts cost 16 cents, } of a pound will cost 
J of 16 cents, or 4 cents. 

2. What is the cost of ^ of a dozen oranges at 30 cents a 
dozen ? 

3. A boy had 90 cents, and spent ^ of it ; how much did 
he spend? 

4. If my wages are 60 dollars a month, how much are they 
for jl^ of a month ? 

5. If 5 boys earn 40 dollars, how much does each earn ? 

6. A cow cost $35 and a sheep ^f as much ; how much did 
the sheep cost ? 

7. How much will | of a bushel of potatoes cost at 60 
cents a bushel? 

Solution. — If a bushel of potatoes cost 60 cents, J of a bushel will 
cost J of 60 cents, or 15 cents ; and f of a bushel will cost 3 times 15 
cents, or 45 cents. 

8. How many eggs in | of a dozen ? 

9. Mr. Ellis is 40 years old and his son is |^ as old ; how 
old is his son? 

10. A boy had 60 cents, but spent f of it ; how much has 
he remaining? 



FRACTIONS. 65 

11. What is the cost of | of an acre of land at 72 dollars 
an acre ? 

12. If a horse cost 120 dollars and a cow -f^ as much, how 
much does the cow cost ? 

13. If 3 calves cost f of 30 dollars, how much does 1 calf 
cost? 

14. ^ of 21 dollars is twice the cost of a pair of shoes ; 
how much did the shoes cost? 

15. A horse cost $150 and a wagon | as much ; how much 
did the wagon cost ? 

16. I of 36 are how many times 3 ? 

17. f of 49 are how many times ^f of 49 ? 

18. 3^ of 88 are how many times f of 40 ? 

19. I of 54 are how many times | of 30? 

20. f of 96 are how many times f of 50 ? 

21. What is i of 8? 

Analysis. — Since ^ of 1 is ^, J of 8 is 8 times J, or f . 

22. What is ^ of 4? ^of 5? 

23. What is i of 5? } of 5? 

24. What is I of 6? |of 7? 

25. What is I of 6? ^ of 8? 

26. What is ^of 8? i of 10? 

27. What is I of 5? 

Analysis. — Since ^ of 5 is f, | of 5 is 2 times J, or y*-. 

28. What is I of 4? | of 7? 

29. What is I of 5? | of 9? 

30. What is I of 6 ? I of 10 ? 

31. What is I of 11? I of 16? 

32. What is I of 18 ? | of 20 ? 

33. What is f of 10? f of 12? 

34. What is f of $8 ? f of 15 bushels ? 

35. When apples are worth $3 a barrel, how much will 3^ 
barrels cost? 

5 



66 MENTAL ARITHMETIC. 

36. When wheat is worth 80 cents a bushel, what is the 
cost of 2| bushels? 

87. When coal is $5 a ton, how much will 4| tons cost ? 

38. If eggs are worth 30 cents a dozen, how much will 3| 
dozen cost ? 

39. When cheese is worth 18 cents a pound, what is the 
cost of 2| pounds? 

40. If land is worth $50 an acre, how much will 5^ acres 
cost? 

41. What is the price of 1 dozen and 5 eggs at 18 cents a 
dozen ? 

42. If a horse travel 8 miles an hour, how far will he 
travel in 5| hours? 

43. What is the cost of 6| bushels of potatoes at 50 cents 
a bushel? 

44. If a family use a barrel of flour in 9 weeks, how long 
will 2| barrels last them ? 

45. What is the cost of 6J pounds of coffee at 30 cents a 
pound ? 

46. When flour is worth $6 a barrel, how much is ^ of a 
barrel worth? 

47. When hay is worth $16 a ton, how much is f of a ton 
worth ? 

48. If 8 bushels of corn cost $4, how much will 3 bushels 
cost ? 

49. What is the cost of | of a yard of linen worth 60 cents 
a yard ? 

50. If I pay 20 cents a dozen for oranges, how much is 
that apiece? 

51. How much will 3 dozen and 5 eggs cost at 18 cents a 
dozen ? 



FRACTIONS. 67 

3. 

Fractions Multiplied by Fractions. 

1. What is I of I? 

Solution. — J of ^ is ^V 5 J of | is 2 times j^^, or ^^ ; and f of | is 2 
times -f^j or ■^^. 

2. What is I of i ? 6. What is | of f? 

3. What is f of I ? 7. What is | of /^ ? 

4. What is I of I? 8. What is | of | ? 

5. What is ^ of i? 9. What is | of |? 

10. What is I of 61 ? 

Suggestion. — 6 J is -\^, hence f of 6 J is f of ^^-. 

11. What is I of 4|? 

12. What is I of 41^? I of 21? 

13. How much will | of a bushel of wheat cost at | of a 
dollar a bushel? 

14. A boy, having i of a pie, gave away ^ of it; what 
part of a pie did he give away ? 

15. If a yard of cloth is worth | of a dollar, how much is 
I of a yard worth ? 

16. What is the cost of | of a pound of butter at § of a 
dollar a pound ? 

17. If I have | of a dollar and give away f of it, what 
part of a dollar do I have remaining? 

1 8. How much can a boy earn in 2| days at | of a dollar 
a day? 

19. At $5| a ton, how much will | of a ton of coal cost? 

20. If a man earn S3^ a day, how much does he earn in 
I of a day ? 

21. How much will 2| dozen of eggs cost at 12^ cents a 
dozen ? 

22. What is the cost of 3| pounds of steak at 18 cents a 
pound ? 



68 MENTAL ARITHMETIC. 

23. When cloth is worth $3^ a yard, what is the cost of 2\ 
yards ? 

24. A man had $7^ and spent | of it for a hat ; what did 
the hat cost? 

25. How much will lOj pounds of sugar cost at 6^ cents a 
pound ? 

26. When pork sells at 12 J cents a pound, what is the cost 
of 5 J pounds? 

27. When eggs are worth 15 cents a dozen, how much do 
3 dozen and 3 eggs cost ? 

28. When meat is selling at 2 pounds for 25 cents, how 
much are 6 J pounds worth? 

What is the product — 

29. Of I by I? 32. Of i of 6 by |? 

30. Of ^ by I ? 33. Of I of 5 by ^ of 7? 

31. Of I of ^ by I ? 34. Of I of 4 by f of 8? 



DIVISION OF FRACTIONS. 

1. 

Fractions Divided by Integers. 

1. If 3 boys earn y^^j of a dollar, how much does 1 boy 
earn? 

2. If 4 hens cost -^-^ of a dollar, how much does one hen cost? 

3. If 2 ducks cost | of a dollar, how much does 1 cost? 

4. If 4 slates cost | of a dollar, how much will one slate 
cost? 

Solution. — If 4 slates cost f of a dollar, 1 slate will cost J of | of a 
dollar, or /^ of a dollar. 

5. If 5 dozen eggs cost | of a dollar, how much does 1 
dozen cost? 

6. When 6 yards of calico are worth ^^ of a dollar, what 
is 1 yard worth? 



FRACTIONS. (jy 

7. If 6 men can do | of a piece of work, how much can 
one man do? 

8. Divide | into 5 equal parts. 

9. When 4 pounds of butter are worth $1^, how much does 
one pound cost? 

10. If 5 men earn $8^, how much does 1 man earn ? 

11. A boy, having 6 quarts of cherries, shared them with 7 
other boys ; what part of a quart did each receive ? 

12. Henry, having 6 oranges, divided them among 4 of his 
companions; how many did each companion receive? 

13. If a boy walk 13J miles in 4 hours, how far does he 
walk in 1 hour? 

14. If 5 yards of cloth cost 12^ dollars, how much will 1 
yard cost? 

15. James, having gathered 12 quarts of berries, gave | 
of them to his mother and sold the remainder for ^ of a dol- 
lar ; how much per quart did he get for them ? 

16. If 6 men in a certain time earn 13^ dollars, how much 
does 1 man earn ? 

17. How many times is 2 contained in | ? In | ? 

18. How often is 5 contained in | ? In f ? 

19. Divide j\ by 4 ; f by 10 ; I by 5. 

20. Divide I by 6 ; If by 5 ; ^f by 8. 

21. Divide 6^ by 3 ; 3^ by 4 ; 5 J by 5. 

22. Divide 2| by 7 ; 6| by 3 ; 14f by 6. 



2. 

Integers Divided by Fractions. 

1. At I of a dollar a bushel, how much corn can be bought 
for $6? 

Solution.— If 1 bushel of corn cost | of a dollar, for $6, or 1^^, as 
many bushels can be bought as f is contained times in \*-, or 8 bushels. 



70 MENTAL ARITHMETIC. 

2. How many hooks at f of a dollar each can I huy 
for $4. 

3. How many hens at -f of a dollar each can I get for $6 ? 

4. If I earn | of a dollar a day, how long will it take to 
earn $9? 

5. How many strips of carpet | of a yard wide will it re- 
quire to cover a floor 6 yards wide ? 

6. How many baskets of peaches at | of a dollar a basket 
can be bought for $8 ? 

7. If it require | of a yard of linen to make an apron, how 
many aprons can be made from 6 yards ? 

8. How many hens at | of a dollar each can be bought 
for $10? 

9. How long will it take a man to earn $15, if he earn $2^ 
a day? 

10. If it require | of a yard of ribbon to make a badge, 
how many badges can be made from 12 yards of ribbon ? 

11. At $1\ each how many turkeys can be bought for $10? 

12. James can earn $1| a day ; how long will it take him 
to earn $14? 

13. How many tons of coal at $5 J a ton can be bought 
for $21? 

14. If a boy spend $1^ a week, in how many weeks will he 
spend $12? 

15. How many pounds of sugar at 6^ cents a pound can 
be bought for 50 cents ? 

16. How oflen is ^ contained in 2 ? In 4 ? In 5 ? 

17. How often is | contained in 3? In 9? In 12? 

18. Divide 4 by f ; 6 by f ; 8 by |. 

19. Divide 4 by f ; 5 by | ; 6 by |. 

20. Divide 5 by |; 8 by |; 10 by f 



FRACTIONS. 71 

3. 

Fractions Divided by Fractions. 

1. If a duck cost f of a dollar, how many ducks can I 
buy for $lf ? 

Solution.— If. 1 duck cost f of a dollar, for $lf , or | dollars, I can 
buy as many ducks as | is contained times in f , or 4 ducks. 

2. How many dozen oranges at ^ of a dollar a dozen can 
I buy for ^ of a dollar ? 

3. How many melons at ^ of a dollar each can I buy 
for $21? 

4. How many bananas at f of a dime each can I get for 
6 dimes? 

5. When paper is ^ of a dollar a quire, how many quires 
can I buy for S3? 

6. At $1^ a yard how many yards of silk can I buy for i 
of a dollar? 

7. If 2 oranges cost a dime, how many can I get for H 
dimes ? 

8. At Y% of a dollar a yard how many yards of cashmere 
can I get for $4|? 

9. If 3 turkeys cost S2|, how many can be bought 
for $7^? 

10. Harry gave 12^ quarts of chestnuts worth 10 cents a 
quart for muslin at 12^ cents a yard ; how many yards did 
he get? 

11. How many bushels of potatoes at | of a dollar a bushel 
can be bought for $4^ ? 

12. At I of a dollar a pound how many pounds of coffee 
can be bought for | of a dollar ? 

13. How many times is | contained in | ? 

Solution 1. — f equals |^, and f equals |f ; ^^ is contained in Jf as 
often as 15 is contained in 16, or Ij^ times. 



72 MENTAL ARITHMETIC. 

Solution 2. — Since 1 is contained in f f times, J is contained in 4 
4 times | times, or V- times, and f is contained in 1 1 of ^^ times, or 
If times, equal to lyV times. 

14. How often is | contained in | ? In f ? 

15. How often is | contained in | ? In | ? 

16. How often is | contained in | ? In |? 

17. How often is | contained in f ? In -j^g ? 

18. How often is | contained in 1^ ? In 2| ? 
Divide — 

19. I by i; ,9, by f. 

20. I by f ; f by |. 

21. 2^ by I; 3^ by §. 

22. 6i by 2^ ;4i by 3^. 

23. 4|by3i; 9^ by 3J. 

24. ^ of I by i ; I of 2| by |. 

25. ^of 3|by|; | of | by | of |. 

26. At $3^ apiece how many sheep can be bought for 
«19i? 

27. If 3 baskets of peaches are worth $2, how many bas- 
kets are worth $2| ? 

28. A farmer divided his land, 85^ acres, among his 3 sons ; 
how much did each get ? 

29. If a boy can run |^ of a mile in a minute, how long will 
it take him to run | of a mile ? 

30. If I buy 2^ bushels of potatoes for $1, how much will 
6| bushels cost ? 

31. If a turkey cost $lf, how many turkeys at the same 
rate can be bought for $8| ? 

32. If a man can do a piece of work in 8^ days, how many 
men will be required to do the work in 2| days ? 

33. How many quarts of berries at ^^ of a dollar a quart 
can be bought for 3^ bushels of wheat worth | of a dollar a 
bushel ? 

34. A woman exchanged 6^ dozen of eggs at J of a dollar 



FRACTIONS. 73 

a dozen for coffee at f of a dollar a pound ; how much coffee 
did she get ? 

35. A farmer sold 22^ pounds of butter at -f^ of a dollar 
a pound, and took in exchange cloth at | of a dollar a yard ; 
how much cloth did he get ? 



COMPLEX FRACTIONS. 

A Complex Fraction is an indicated division, in which 
either the numerator or the denominator or both are fractions. 

5 

I. What is the value of ^ ? 

Solution. — The fraction -^ is equivalent to f divided by 3 J, or -i/-. 

Since f equals |f and -i/- equals |f, | divided by 3 J is equal to 16 di- 
vided by 96, or I. 

What is the value — 

2. Off? 6. Of ^? 

i H 

3. Off? 7. Of^? 

3 3 

4. Off? 8. Offf? 

5. Of ^? 9. Of ^^p? 

10. A man's wages are $2| a day ; how long does it take 
him to earn 813| ? 

II. If 4^ pounds of meat cost 49 J cents, how much does 
it cost per pound ? 

12. If I of I of a ton of coal cost $1|, how much will a 
ton cost ? 



74 MENTAL ARITHMETIC. 

RELATION OF NUMBERS. 

1. 

Relation of Integers to Integers. 

1. If 2 apples cost 3 cents, how much will 10 apples cost ? 

Solution. — If 2 apples cost 3 cents, 10 apples, or 5 times 2 apples, 
will cost 5 times 3 cents, or 15 cents. 

2. If 2 bananas cost 5 cents, how much will 6 bananas cost ? 

3. If 3 pencils cost 10 cents, how much will 12 pencils cost? 

4. If 5 boys earn $3, how much will 20 boys earn in the 
same time ? 

5. What is the relation of 30 to 5? 
Ans. — 30 is 6 times 5. 

What is the relation — 

6. Of 6 to 3? Of 6 to 2? Of 12 to 4? 

7. Of 10 to 5? Of 20 to 5? Of 20 to 4? 

8. Of 15 to 3? Of 15 to 5? Of 16 to 4? 

9. Of 30 to 6? Of 40 to 8? Of 36 to 6? 

10. Of 50 to 25? Of 60 to 12? Of 12 to 6? 

11. If $20 buy three sheep, how many may be bought for 
$60? 

12. If 20 cents buy a dozen of eggs, how many eggs will 
30 cents buy ? 

13. If a man can walk 10 miles in 3 hours, how far can he 
walk in 9 hours ? 

14. If 10 cents buy 4 oranges, how many oranges will 5 
cents buy? 

15. If 5 horses cost $750, how much will 10 horses cost? 

16. What is the cost of 3 cows if 9 cows cost $360 ? 

17. A merchant sold shoes at the rate of 2 pairs for S7; 
how many pairs can I get at the same rate for $21 ? 

18. If $15 buy 12 turkeys, how many will $5 buj? 



FRACTIONS. 75 

2. 

Relation of Fractions to Integers. 

1. What part of 6 is | ? 

Solution.— 1 is i of 6, and f is f of i of 6, or f^ of 6. 

2. What part of 5 is | ? | ? | ? 

3. What part of 4 is | ? | ? | ? 

4. What part of 6 is | ? |? f? 

5. What part of 8 is 1^ ? 3i ? 5^ ? 

6. What is the relation of | to 20 ? 

7. What is the relation of § to 12? 

8. What is the relation of | to 16? 

9. What is the relation of 3^ to 14? 
10. What is the relation of 2i to 6 ? 



3. 

Relation of Integers to Fractions. 

1. What is the relation of 4 to -^? 
Solution.— 4, which is -^Z, is 5 times f . 

What is the relation — 

2. Of 6 to f? 6. Of 12 to f? 10. Of 3| to I? 

3. Of 8 to I? 7. Of 12 to -I? 11. Of 6i to 1^2? 

4. Of 12 to I? 8. Of 12 to 4? 12. Of 30 to 3|? 

5. Of 9 to I? 9. Of 10 to I? 13. Of 16 to 3i? 



4. 

Relation of Fractions to Fractions. 

1. What part of f is | ? 

Solution. — Since |^ is | of |, f, or 1, is 6 times J of |, or f of f ; 
and since 1 is f of |, f is | of f of |, or |f of f . 



76 MENTAL ARITHMETIC. 

What part — 



2. 


Of ^ is i ? 


6. Of |is|? 


10. 


Off is2i? 


3. 


Oflisf? 


7. Of lis f? 


11. 


Of 61 is 1? 


4. 


Of lis 1? 


8. Of S is 1 ? 


12. 


Of 8| is 1 ? 


5. 


Offisf? 


9. Of lis 4? 


13. 


Of 12^ is 6| ? 


14. 


What is the relation of 3^ to 16 


V 




15. 


What is the relation of 31 1 to 6 


•1? 





ANALYSIS, OR COMPARISON OF NUMBERS. 

1. 

Integers Compared with Integers. 

1. If 5 boys earn SI 2, how much will 3 boys earn ? 

Analysis. — Since 5 boys earn $12, one boy earns J of $12, or ^/ dol- 
lars; and 3 boys earn 3 times J/ dollars, or Y" dollars, equal to 7 J 
dollars. 

2. If 5 barrels of flour cost $30, how much will 4 barrels 
cost? 

3. If a horse travel 36 miles in 6 hours, how far does he 
travel in 2 hours, at the same rate ? 

4. If 5 yards of ribbon cost 55 cents, how much will 12 
yards cost? 

5. If 3 dozen eggs cost 60 cents, how much will 5 dozen 
cost? 

6. When 6 yards of muslin cost 48 cents, how much do 10 
yards cost? 

7. If 10 quarts of milk are worth 60 cents, how much are 
3 quarts worth ? 

8. George can earn 80 dollars in 5 weeks ; how much can 
he earn in 3 weeks ? 

9. If 5 weeks' board cost $20, how much do 8 weeks* 
board cost? 



FRACTIONS. 77 

10. If 5 tons of coal are worth $26, how much are 3 tons 
worth ? 

11. If 4 quarts of oil cost 60 cents, how many quarts can 
I buy for 75 cents? 

12. What is the cost of 8 dozen of eggs, if 5 dozen cost 
70 cents? 

13. If a coach travels 40 miles in 8 hours, how far does it 
travel in 10 hours? 

14. How much will 15 cows cost if 5 cost $75 ? 

15. If 20 brooms cost $5, how many brooms can be bought 
for $12? 

16. If I exchange 10 yards of silk at 90 cents a yard for 
butter, how much butter do I get if 4 pounds of butter are 
worth $1.20? 

17. At the rate of 10 cents a dozen how many apples can 
I get for 25 cents ? 

18. A farmer sells 15 hens worth 40 cents each, and takes 
in exchange coffee worth 30 cents a pound ; how many pounds 
of coffee does he get ? 

19. If 4 men can do a piece of work in 6 days, how long 
will it take 12 men to do the same work? 

Analysis. — If 4 men can do a piece of work in 6 days, it will take 
1 man 4 times 6 days, or 24 days, to do the work, and 12 men will do 
it in jV of 24 days, or 2 days. 

20. How long will it take 1 man to do a piece of work if 
6 men do the same work in 3 days ? 

21. If 6 men do a piece of work in 4 days, how many men 
will be required to do the same work in 1 day ? 

22. If 8 men can dig a cellar in 5 days, how long will it 
take 4 men to dig it? 5 men? 10 men? 

23. If 5 men can build a wall in 12 days, how many men 
will it take to build it in 1 day ? In 6 days ? In 10 days ? 

24. A laborer agrees to work 4 months for 60 dollars ; if 
he work 6 months, how much does he receive? 



78 MENTAL ARITHMETIC. 

25. If 3 men do the carpenter work on a house in 20 days, 
how long would it have taken 5 men to do it ? 

26. If 4 men can build a stone wall in 15 days, how many 
men could have built it in 12 days ? In 5 days ? In 6 days? 



2. 

Comparison of Fractions with Integers. 

1. If j\ of a dozen eggs cost 14 cents, how much will 4 
dozen cost? 

Analysis. — If ^^ of a dozen of eggs cost 14 cents, t\ of a dozen will 
cost ^ of 14 cents, or 2 cents ; ||, or 1 dozen, will cost 12 times 2 cents, 
or 24 cents ; and 4 dozen will cost 4 times 24 cents, or 96 cents. 

2. If f of the price of a book is 30 cents, what is the full 
price ? 

3. How old am I if I of my age is 24 years ? 

4. If $30 is I of what a man paid for a cow, how much did 
she cost ? 

5. When 2^ pounds of butter cost 90 cents, what is the 
cost of 6 pounds? 

6. Mr. A paid $150 for a horse, and a cow cost | as much ; 
how much did the cow cost? 

7. What is the cost of 10 sheep if | of 8 dollars is the cost 
of 1 sheep ? 

8. I bought some grain for $120, which is | of what I sold 
it for ; how much did I gain ? 

9. George earned $15, which is | of what he already had ; 
how much had he at first ? 

10. Henry lost 12 marbles, which is f of his number at 
first ; how many has he remaining ? 

11. George ate 8 plums, which is f of the number he has 
remaining ; how many had he at first ? 

12. 9 is J of what number? 



FRACTIONS. 79 

13. 8 is I of what number ? 

14. 12 is f of what number? 

15. 20 is I of what number? 

16. 18 is I of what number? 

17. 14 is 5 of what number? 

18. 15 is I of what number? 

19. If 16 is I of a number, what is 3 times the number? 

20. If 18 is I of a number, what is 5 times the number? 

21. 12 is ^ of I of what number? 

22. 30 is i of f of what number? 

23. 18 is f of f of what number ? 

24. 20 is f of I of what number? 

25. 24 is I of I of what number? 

26. I of 12 is 4 of what number? 

27. I of 18 is I of what number ? 

28. f of 100 is j% of what number? 

29. I of 18 is I of what number? 

30. ^ of 28 is f of what number? 

31. A hat cost $6, which is f the cost of a coat ; how much 
did the coat cost ? 

32. George is 12 years old, which is | of ^ of his father's 
age ; how old is the father ? 

33. The cost of a sleigh is $60, which is | of | the cost of 
a carriage ; what is the cost of the carriage ? 

34. A boy has 20 hens, and this is f of | of his brother's 
number ; how many has the brother ? 

35. A watch is worth $75, which is | of 5 times what a 
chain cost ; what is the cost of the chain ? 

36. If a wagon is sold for $100, which is | of 2 times its 
cost, how much is gained ? 

37. A man sold his house for $2400, which was | of 6 times 
what it cost him ; how much did he gain on the sale ? 



80 mental arithmetic. 

Comparison of Integers with Fractions. 

1. If 6 yards of muslin cost 48 cents, how much will 1^ 
yards cost? 

Analysis. — If 6 yards of muslin cost 48 cents, 1 yard will cost J of 
48 cents, or 8 cents ; and 1 J, or }, yards will cost J of 8 cents, or 10 cents. 

2. If 3 tons of coal cost SI 5, how much will | of a ton 
cost? 

3. A cow cost $36 and some sheep f as much ; how much 
did the sheep cost ? 

4. If a dozen eggs cost 24 cents, how much will | of a 
dozen cost? 

5. When eggs are worth 30 cents a dozen, how much will 
2| dozen cost? 

6. If 8 acres of land produce 240 bushels of wheat, how 
many bushels will 3J acres produce ? 

7. A farmer had 60 sheep ; if he sell ^ of them, how many 
will he have remaining ? 

8. If 3 yards of cloth cost $9, how much will | of a yard 
cost? 

9. When 4 pounds of tea cost $3, how much will 5^ 
pounds cost? 

10. If 6 bushels of oats cost $2, how much will 2| bushels 
cost? 

11. Henry earned $60 in 10 weeks ; how much could he 
earn in 4^ weeks ? 

12. When 7 bushels of corn cost $5|, how much will 10| 
bushels cost ? 

13. When 2| pounds of cheese cost 44 cents, how much 
will 3^5 pounds cost? 

14. If I of 4 dozen of eggs cost f of a dollar, how much 
will 2 J dozen cost? 



FRACTIONS. 81 

15. What number is that to which, if | of itself be added, 
the result will be 56? 

16. Henry sold f of his hens and had 35 remaining ; how 
many had he at first ? 

17. If f of 14 yards of silk cost $12, how much will 3| 
yards cost? 

18. If "I" of 10 cords of wood cost $6, how^ much will | of 
3 cords cost? 

19. If 2 J barrels of apples cost S9, how much will 3| bar- 
rels cost ? 

20. When 40 acres of land cost $1800, how much will 5| 
acres cost ? 



4. 

Comparison of Fractions with Fractions. 

1. If i of a bushel of wheat cost 64 cents, how much will 
I of a bushel cost ? 

Analysis. — If | of a bushel of wheat cost 64 cents, ^ of a bushel 
will cost I of 64 cents, or 32 cents ; and 1 bushel will cost 3 times 32 
cents, or 96 cents. Since 1 bushel of wheat costs 96 cents, f of a bushel 
will cost f of 96 cents, or 72 cents. 

2. I of my age is 30 years ; what is | of my age ? 

3. If I of a pound of cheese cost 6 cents, how^ much will | 
of a pound cost ? 

4. When f of my monthly salary is $9, how much do I 
earn in | of a month ? 

5. I of the cost of my horse is $100 ; what do I get for him 
if I sell him at f of his cost ? 

6. When | of a dozen of eggs cost 18 cents, how much will 
1| dozen cost? 

7. If 1^ barrels of apples cost $6, what must I pay for 2| 
barrels ? 



82 MENTAL ARITHMETIC. 

8. If ^ of a box of soap cost 90 cents, what is the cost of 
21 boxes? 

9. If a horse travel 14 miles iu 2^ hours, how far will he 
travel in 5^ hours? 

10. If 3^ pounds of meat cost 50 cents, how much will 4 J 
pounds cost? 

11. If 1^ of a ton of coal is worth $3|, how much are 2| 
tons worth ? 

12. How much will o^ pounds of sugar cost, if 2^ pounds 
cost 20 cents ? 

13. When 2| barrels of flour are worth $13^, how much 
are 1^ barrels worth ? 

14. 12 years is | of my age; what is twice my age? 

15. If 3| pounds of butter are worth $1.35, what are 2i 
pounds worth ? 

16. If I of a barrel of vinegar is worth S3|, how much 
are If barrels worth ? 

17. How much will 1| yards of silk cost, if | of a yard 
cost I of a dollar ? 

18. If a ship sail 4^ miles in § of an hour, how far will it 
sail in 8 hours ? 

19. Henry can mow an acre of grass in 10^ hours, and 
William in | of that time ; how long does it take William to 
mow the acre ? 

20. When 3J dozen of eggs are worth 70 cents, how much 
are 3^ dozen worth ? 

21. If 2 yards of cloth cost $3 J, how many yards can I 
buy for $5 ? 

22. If a boy can earn $64 in 4 days, how much can he earn 
in I of |- of a day ? 

23. How many sheep at $5i apiece can I get for 5 tons of 
coal at $^ a ton ? 

24. How many pounds of butter at j% of a dollar a pound 
will pay for 16 pairs of chickens at | of a dollar a pair? 



FRACTIONS. 83 

25. How many pounds of coffee at y% of a dollar a pound 
can I get for 3 pairs of ducks at | of a dollar a pair ? 



A Fractional Part with an Integral Remainder. 

1. A boy lost I of his money and had $12 remaining; how 
much had he at first ? 

Analysis. — Since he lost f of his money, he had remaining \ — f, 
or f, of his money, which is |12: if f of his money is $12, \ of his 
m-oney is \ of $12, or $3, and ^, or his money, is 7 times $3, or $21. 

2. George spent | of his money and had $20 remaining ; 
how much had he at first ? 

3. A farmer sold ^ of his wheat to one man and \ to 
another, and had 160 bushels remaining ; how much had he 
at first? 

4. Mr. A sold | of his corn to one man and \ to another, 
and has 210 bushels remaining; how much had he at first? 

5. If Henry spend ^ of | of his money and has $30 re- 
maining, how much had he at first ? 

6. In an orchard ^ of the trees are peach trees, \ pear 
trees, and \ apple trees; the remainder, 24, are plum trees; 
how many are there of each, and how many together? 

7. Samuel spent J of hi& money, then earned \ as much as 
he spent, and had $20 ; how much had he at first? 

8. A lady sold -J of her hens, then bought | as many as she 
had at first, and then had 84 ; how many had she at first ? 

9. George won 40 marbles, then lost f of what he won, and 
had 25 ; how many had he at first ? 

10. ^ of a man's money, being increased by \ and \ of his 
money, equals $620 ; how much money has he? 

11. Henry spent | of his money, and then earned | as 



84 MENTAL ARITHMETIC. 

much as he had at first, and then had $65 ; how much had 
he at first ? 

12. My expenses are -| of my annual income; what is my 
income, if I save $340 ? 

13. A man's board is J of his salary, and his other expenses 
§ ; what is his salary, if he save $200 ? 

14. What number diminished by its f and its j equals 48 ? 

15. A merchant who owed me failed, and I received only 
I of my money ; what was the debt, if I received $540 ? 

16. I find that :j of ^ of the trees in my orchard have died, 
but I still have 76 ; how many had I at first ? 

17. If James spends } of his income for board and | for 
other expenses, what is the income, if he saves $120? 

18. If I pay ^ of my salary for rent and ^ of | of it for 
other expenses, what is the salary, if I have remaining $210? 

19. After spending |, |, and I of my money I find $26 is 
^ of the remainder ; how much money had I ? 

20. Mr. Thomas sold ^ of his potatoes to A, | of the re- 
mainder to B, and had 40 bushels remaining; how many 
bushels had he at first? 

21. What number is that to which if you add | of itself 
the result is 140? 

22. A young man, being asked his age, replied. If to my 
age you add ^ and | of my age, the sum is 39 years ; how old 
was he ? 

23. Paid | of my money for a book, I for pens, and 26 
cents for a slate, and then had ^ of my money remaining ; 
how much had I at first? 

24. I spent h of my money, then borrowed | as much as I 
had remaining, when I had 20 cents less than at first ; how 
much hud I at first ? 

25. I of 12 is how many eighths of 16? 

26. I of 30 is how many fifths of 25? 

27. I of 48 is how many sevenths of 49? 



FRACTIONS. 85 

28. ^% of 80 is how many sixths of 36 ? 

29. f of 72 is I of 10 times what number? 

30. I of 80 is f of 20 times what number ? 

31. f of 56 is I of 5 times what number? 

32. j\ of 88 is I of 12 times what number? 

33. I of 60 is I of how many times 5 ? 

34. I of 64 is I of how many times 14? 

35. I of 48 is ^ of how many times 7 ? 

36. I of 40 is f of how many times | of 18? 

37. I of 72 is I of how many times | of 25? 

38. I of I of 60 is what part of 84? 

39. I of f of 28 is what part of 48 ? 

40. I of f§ of 40 is what part of | of 80? 

41. I of ^of 32 is what part of | of 120? 

42. f of Robert's money equals | of William's, and | of 
William's is | of George's ; how much money has each, if 
George has $48? , 



Review Problems. 

1. Henry had | of a dollar and spent -J- of a dollar ; how 
much had he remaining ? 

2. If I be added to a certain fraction, the sum will be | ; 
what is the fraction ? 

3. If I of a dozen of oranges cost 16 cents, how much will 
3A dozen cost? 

4. Bought I of a ton of coal in June, | of a ton in July, 
and 1| tons in August ; how much did I buy in the three 
months ? 

5. Bought a coat for $12| and a hat for $4| ; how much 
change did I get from a 20-dollar bill? 

6. Henry is 20 years old, and | of his age is |- of his 
father's age; how old is his father? 



86 MENTAL ARITHMETIC. 

7. If to my age you add | and | of my age, the sum will 
equal 50 years ; what is my age ? 

8. If 3 men can do a piece of work in 10 days, how long 
will it take 5 men to do the same work ? 

9. If 4 men do a piece of work in 12 days, how many men 
will be required to do it in 8 days? 

10. A watch and a chain cost $140 ; how much does each 
cost if the chain costs f as much as the watch ? 

11. A boy sold a ball for 60 cents, which was ^ less than it 
cost him ; how much did it cost him ? 

12. A merchant's expenses were $1020 : of this he paid | 
for insurance, ^ for interest, and the remainder for rent ; how 
much did he pay for each ? 

13. Three houses together cost $29,000 : the first cost | as 
much as the second, and the second | as much as the third ; 
how much did each cost? 

1 4. Having a journey of 600 miles to perform, I traveled § 
the distance the first day, J the second day, and the remain- 
der the third day ; how far did I travel the third day ? 

15. I spent 4 of my money for board and | for traveling 
expenses ; if I had $44 left, how much had I at first ? 

16. If a bushel of corn cost | of a dollar and a bushel of 
wheat I of a dollar, what is the difference in the cost of 6 
bushels of each ? 

17. How much more will 2 dozen of oranges at | of a dol- 
lar a dozen cost than 3 dozen of apples at 2 for 3 cents ? 

18. A boy is f as old as his father and ^ as old as his 
mother ; how old is each, if the boy is 16 years of age ? 

19. I buy turkeys at the rate of 3 for $5 : if I sell them 
at $1.75 each, how much do I make on four? 

20. George is 12 years old : if 4 years be added to his age, 
the sum will be ^j- of his father's age ; how old is the father ? 

21. Iff of I of the cost of a cow is $32, and I sell her for 
I of f of $81, how much do I make? 



FRACTIONS. 87 

22. I have $75 : I buy 40 turkeys at the rate of 6 for $4 ; 
how much money have I remaining? 

23. If a boy runs 15-^ miles in 2-h hours, how far does he 
run in an hour ? 

24. I have $20 : if I buy books at $3^ a dozen and have 
$6 left, how many books do I buy ? 

25. George has $2f and Henry has $3] ; if they share their 
money equally, how much has each ? 

26. A man gave | of his money for a cow, then bought 
some fowls for $6, and had \ of his money remaining ; how 
much had he at first ? 

27. A has $24-|, B has $5| more than A, and C has $10| 
less than A and B together ; how many dollars have B and 
C each ? 

28. A boy bought oranges at the rate of 15 cents a dozen 
and sold them at the rate of two for five cents ; how much 
did he make on each orange? How much on a dozen? 

29. If by selling a house for $1800 I lose J of its value, 
what was the house worth? 

30. If by selling a horse for $160 I gain | of the cost, 
what did he cost? 

31. A boy is { as old as his father and | as old as his 
mother; how old is each if the boy is 15 years of age? 

32. How many barrels of flour at $5J a barrel can I get 
for 35 bushels of corn at | of a dollar a bushel ? 

33. If I can earn $5{ in 2 J days, how much can I earn in 
a day? 

34. If I were 3 times as old as I am now, f of my age 
would be 24 years ; how old am I ? 

35. If I of the sum of two numbers is 27, what are the 
numbers if one is twice as great as the other? 

36. Henry is 20 years old : if his age were increased f , the 
sum would be 2| times his brother's age; how old is his 
brother ? 



88 MENTAL ARITHMETIC. • 

37. If 6 men can do a piece of work in 5 days, how much 
of the work can 3 men do in 8 days ? 

38. How many ducks can I buy for four dollars at the rate 
of five ducks for two dollars ? 

39. If a man can earn $40 in 1 J months, how much can 
27 men earn in 2jJ months? 

40. I sold my house for | of the cost, and lost $1500; 
how much would I have made if I had sold it for 1^ times 
the cost? 

41. I bought a horse for $87^ and sold him at a gain of 
•^ of the cost ; how much did I get for him ? 

42. What number added to 3 times f of 16 will equal 
26? 

43. I of I of 60 is how many times | of 7 J times 6 ? 



DENOMINATE NUMBERS. 

A Denominate Number is one whose unit is named ; as 2 
feet, 6 pounds, etc. 

Numbers are of the same denomination when they have 
the same unit ; as, 3 inches, 6 inches, etc. 

A Compound Number is one consisting of two or more de- 
nominations of the same nature ; as, 2 yards, 2 feet, 3 inches. 



MEASUKES. 



A Measure is a standard unit used in estimating. 
A Quantity is measured by finding how often it contains 
the standard unit of measurement. 

Note. — United States Money has been treated of on pages 39-41. 



DENOMINATE NUMBERS. 89 

ENGLISH OE STEELING MONEY. 
English or Sterling Money is the currency used in England. 

Table. 

4 farthings (qr.) = 1 penny, d. 

12 pence = 1 shilling, s. 

20 shillings = 1 pound, £. 

21 shillings = 1 guinea. G. 

1. How many farthings in 2 pence ? In 3 pence ? In 4 d.? 
In5d.? InSd.? InlOd.? 

2. How many pence in 8 farthings? In 12 farthings? In 
20 farthings ? In 60 farthings ? 

3. How many pence in 3 shillings ? In 4 s. ? In 5 s. ? 
In 7 s.? In 10 s.? 

4. How many shillings in 36 pence ? In 60 d. ? In 24 d. ? 

5. How many shillings in 5 £ ? In 3 £ ? In 6 £ ? 

6. How many pounds in 40 shillings ? In 80 s. ? In 60 s. ? 

7. What part of a pound is 5 shillings ? 

8. What part of 2 shillings is 6 pence ? 



LONG MEASUEE. 

Long Measure is used in measuring length and distance. 
In measuring distance the following table is used: 

Table. 

12 inches (in.) = 1 foot, ft. 

3 feet =1 yard, yd. 

b\ yards, or 16^ feet = 1 rod, rd. 
320 rods, or 5280 feet = 1 mile, mi. 



90 MENTAL ARITHMETIC. 

Note. — The following also are used: 

1 hand, equals 4 inches, used in measuring horses. 

1 size equals J inch, used by shoemakers. 

1 span equals 9 inches, used by sailors. 

1 fathom equals 6 feet, used by sailors. 

1 pace equals 3 feet, used in measuring distances. 

1 league equals 3 miles, used in measuring distances at sea. 

1. How many inches in 3 feet? 5 feet? 8 feet? 

2. How many feet in 24 inches ? 60 inches? 36 inches? 

3. How many feet in 3 yards? 6 yards? 10 yards? 

4. How many yards in 6 feet? 15 feet? 21 feet? 

5. How many yards in 2 rods ? 4 rods ? 5 rods ? 

6. How many rods in 11 yd. ? In 44 yd. ? In 16^ yd. ? 

7. How many rods in 3 miles ? In 5 miles ? 

8. How many miles in 640 rods ? In 960 rods ? 

9. How many inches in 5 yd. ? In 3 yd. ? In 2 yd. ? 

10. How many inches in 2 ft. 5 in. ? In 5 ft. 4 in. ? 

11. How many rods in ^ of a mile? In ^ mi.? 

12. How many inches in 4j ft.? In 6| yd.? 

13. What part of a foot is 2 inches ? 4 in. ? 8 in. ? 

14. What part of a yard is 1 ft. ? 1^ ft. ? 10 in. ? 

15. How many feet and inches high is a horse 16| hands 
high? 

16. How many feet deep is water 5 fathoms deep ? 

17. What is the difference in size between a number 3 shoe 
and a number 5 shoe? 

18. The width of a street is 20 paces ; how many feet wide 
is it ? 

19. How far from land is a ship that is 8 leagues distant? 

20. What part of a foot is ^ inch ? 

21. What part of a yard is ^ inch ? 

22. How many inches in 1^ yards? 

23. How much greater is ^ yard than ^ foot ? 



DENOMINATE NUMBERS. 91 



SURFACE OR SQUARE MEASURE. 

Square Measure is used in measuring areas or surfaces. 

The area of a surface, which has length and breadth only, 
is expressed by the product of the numbers representing 
these two dimensions. 

In measuring surfaces the following table is used : 

Table. 

144 square inches (sq. in.) = 1 square foot, sq. ft. 

9 square feet = 1 square yard, sq. yd. 

301 square yards, or ) ^ , i i t^ 

2721 square feet } ^ ^ ^'1"'^'" '"^ "' ^'''^' "^- '' ^- 
160 square rods — 1 acre, A. 

640 acres = 1 square mile, sq. mi. 

Note. — The rood, equal to 40 square rods, is now but seldom used. 

1. How many square inches in 2 sq. ft. ? In 3 sq. ft. ? 

2. How many square feet in 244 sq. in. ? In 432 sq. in. ? 

3. How many square feet in 2 sq. yd. ? In 5 sq. yd. ? 

4. How many square yards in 27 sq. ft. ? In 54 sq. ft. ? 

5. How many rods in 3 A. ? In 5 A. ? In | A. ? 

6. How many rods in a piece of land 40 rods long and 8 
rods wide ? How many acres ? 

7. How many acres in | sq. mi. ? In | sq. mi. ? 

8. How many acres in a township 6 miles square ? 

9. What is the difference between 4 feet square and 4 
square feet? 

10. How many acres in a plot of ground 24 rods long and 
20 rods wide ? 

11. My garden is 150 feet long and 40 feet wide ; how many 
square feet does it contain, and how many square yards ? 



92 MENTAL ARITHMETIC. 



CUBIC OR SOLID MEASURE. 

Cubic Measure is used in measuring the contents of solids. 
The volume of a body is expressed by the product of its 
three dimensions, length, breadth, and thickness. 
The following is the table used in measuring solids : 

Table. 

1728 cubic inches (cu. in.) = 1 cubic foot, cu. fk. 
27 cubic feet = 1 cubic yard, cu. yd. 

128 cubic feet = 1 cord, C. 

Note. — A cubic yard of earth is called a load. 24| cu. ft. equals a 
perch of stone or masonry. 

1. How many cubic inches in 3 cu. ft. ? In 2 cu. ft. ? 

2. How many cubic feet in 3456 cu. in. ? 

3. How many cubic feet in 2 cu. yd. ? In 4 cu. yd. ? 

4. How many cubic yards in 81 cu. ft. ? In 54 cu. ft. ? 

5. How many cubic feet in 3 perch of stone ? 

6. In a 3-in. cube how many cubic inches ? 

7. How many cubic feet in 5 cords ? 

8. What is the difference between a 4-in. cube and 4 cubic 
inches ? 

9. How many cubic inches in a block 8 in. long, 5 in. deep, 
and 4 in. wide? 

10. How many cubic feet in a wall 10 ft. long, 6 ft. high, 
and 1^ ft. thick? 

11. How many cords in a pile of wood 40 ft. long, 5 ft. 
high, and 4 ft. wide? 

12. How many cubic feet in a piece of timber 16 ft. long, 
1^ ft. wide, and 2 ft. thick ? 

13. How many loads of clay in a cellar 20 ft. wide, 30 ft, 
long, and 4 ft. deep ? 



DENOMINATE NUMBERS. 93 

WEIGHT. 

Avoirdupois Weight. 

Avoirdupois Weight is that used in weighing produce, coal, 
iron, groceries, etc., but not gold, silver, or precious stones. 
The following is the table used : 

TABLE. 

16 ounces (oz.) = 1 pound, lb. 

100 pounds = 1 hundredweight, cwt. 

20 hundredweight, or 2000 pounds ■-= 1 ton, T. 

Notes.— 1. The Avoirdupois pound contains 7000 grains. 
2. In collecting duties the U. S. custom-house officers use what is 
called the long ton (2240 lb.). 

The following units also are used : 

196 lbs. of flour = 1 barrel, bbl. 

200 lbs. of beef or pork = 1 barrel. 
100 lbs. of dry fish = 1 quintal. 

100 lbs. of nails = 1 keg. 

1. How many ounces in 3 lb. ? In 5 lb. ? In 2^ lb. ? 

2. How many pounds in 32 ounces ? In 48 oz. ? 

3. How many pounds in 5 tons ? In 4 tons ? 

4. How many tons in 6000 lb. ? In 4000 lb. ? 

5. How many pounds in 3A tons ? In 4{ tons ? 

6. How many pounds in ^ barrel of flour? In | barrel? 
In I barrel ? 

7. If I sell 2 T. 3 cwt. of straw at 80 cts. a hundredweight, 
and take corn in exchange at 40 cents a bushel, how much 
corn do I get ? 



94 MENTAL ARITHMETIC. 

Troy Weight. 

Troy Weight is used in weighing gold, silver, gems, and pre- 
cious stones. 

The following is the table : 

TABLE. 

24 grains (gr.) = 1 pennyweight, pwt. 
20 pennyweights — 1 ounce, oz. 

12 ounces = 1 pound, lb. 

Note. — The Troy pound is 5760 grains. The Troy ounce is there- 
fore 480 grains, while the Avoirdupois ounce is -}^ of 7000 grains, or 
437^ grains. 

1. How many grains in 2 pwt. ? In 4 pwt. ? 

2. How many pennyweights in 72 gr. ? Id 120 gr. ? 

3. How many pennyweights in 3 oz. ? In 5 oz. ? 

4. How many ounces in 24 pwt. ? In 48 pwt. ? 

5. How many ounces in 4 lb. ? In 6 lb. ? 

6. How many pounds in 36 oz. ? In 60 oz. ? 

7. How much heavier is a pound of iron than a pound of 
gold? 

8. How much heavier is an ounce of gold than an ounce 
of feathers ? 

9. How many ounces in 3 lb. 3 oz. Troy ? 
10. How many spoons, each weighing \\ ounces, can be 
made from a silver bar weighing 2 lb. 3 oz. ? 



CAPACITY. 
Liquid Measure. 



Liquid Measure is used in measuring liquids. 
The following is the table used : 



DENOMINATE NUMBERS. 95 



4 gills (gi.) == 1 pint, pt. 
2 pints = 1 quart, qt. 

4 quarts = 1 gallon, gal. 

Note. — The standard liquid gallon contains 231 cubic inches. 

The barrel and the hogshead are no longer fixed measures, but in 
estimating the capacity of cisterns, reservoirs, etc. 31 1 gallons are al- 
lowed to a barrel, and 7| gallons to a cubic foot. 

1 . How many gills in 2 pints ? In 4 pt. ? In 3| pt. ? 

2. How many pints in 12 gills ? In 16 gi. ? In 60 gi. ? 

3. How many pints in 4 quarts? In 12 qt. ? In 8 J qt. ? 

4. How many quarts in 6 pints? In 10 pt. ? In 16 pt. ? 

5. How many quarts in 3 gallons ? In 5 gal. ? In 4| gal.? 

6. How many gallons in 8 quarts ? In 30 qt. ? In 12 qt. ? 

7. If a pint of vinegar is worth 3 cents, how much is 1 
gallon worth ? 

8. How much will a pint of oil cost at 40 cents a gallon ? 

9. How much will 3 gal. 1 pt. of molasses cost at 20 cents 
a quart? 

10. How many pints in 3^ gallons? 



Dry Measure. 

Dry Measure is used in measuring grain, fruit, salt, soft 
coal, etc. 

The following is the table used : 

TABLE. 

2 pints (pt.) = 1 quart, qt. 
8 quarts = 1 peck, pk. 

4 pecks = 1 bushel, bu. 

Notes. — 1. A pint dry measure equals nearly 1 J pints liquid measure. 
2. In measuring seeds, grain, etc. the measure must be even full. 
A heaped measure is equal to IJ even measures. 



96 MENTAL ARITHMETIC. 

1. How many quarts in 2 pecks? In 5 pk. ? In 4^ pk.? 

2. How many pecks in 16 quarts ? In 40 qt. ? In 34 qt. ? 

3. How many pecks in 3 bushels? In 4^ bu.? In 5 bu. 
Ipk.? 

4. How many bushels in 12 pecks ? In 18 pk. ? In 21 pk. ? 

5. How many quarts in 1 bu. 1 pk. ? In 43 pk. ? 

6. How many quarts in 3 bu. 1 pk. 3 qt. ? 

7. How much will 3| bushels of beans cost at 6 cents a 
quart ? 

8. When potatoes are worth 50 cents a bushel, how much 
are they worth a peck ? 

9. What is the cost of 1 peck 3 quarts of beans at $1.60 a 
bushel? 

10. When chestnuts are worth $2.56 a bushel, how many 
quarts can I get for 80 cents? 



Time. 

T^me Measure is used in measuring time or duration. 
The following is the table : 

table. 

60 seconds (sec.) = 1 minute, min. 

60 minutes = 1 hour, hr. 

24 hours = 1 day, da. 

7 days = 1 week, wk. 

365 days = 1 year, yr. 

366 days = 1 leap year. 
12 calendar months = 1 year. 

100 years = 1 century, C. 

Notes. — 1. In business transactions 30 days are considered a month, 
and 12 months a year. 



DENOMINATE NUMBERS. 97 

2. A. M. denotes before noon ; M., noon ; and P. M., after noon. 

3. The months of April, June, September, and November have 30 
days each. February has 28 days, except in leap year, when it lias 
29. „A11 the other months have 31 days each. 

4. Every year whose number is divisible by 4 is a leap year, except 
the centennial years that are not divisible by 400. 

1. How many seconds in 5 minutes ? In ^ min. ? In 2j 
min. ? 

2. How many minutes in 120 seconds? In 30 sec. ? In 
80 sec? 

3. How many minutes in 4 hours ? In 1| hr. ? In 2 hr. 
7 min. ? 

4. How many hours in 180 minutes ? In 90 mi^. ? In 48 
min.? 

5. How many hours in 3 days ? In 2^ da. ? In 5 da. 
6 hr.? 

6. How many days in 72 hr. ? In 60 hr. ? In 48 hr. ? 

7. How many days in 5 wk. ? In 3 wk. 4 da. ? 

8. How many days from June 1 to July 16 inclusive? 

9. How much time passes between 6 o'clock in the morn- 
ing and half-past 2 in the afternoon? 

10. How long is it from 10 minutes of 8 in the morning to 
12 minutes of 8 in the evening? 

11. How long is it from 15 minutes of 7 in the morning to 
12 minutes past 8 in the evening? 

12. If I give a note June 1 for 90 days, when is it due? 



Angles. 



Circular or Angular Measure is used in measuring angles 
and arcs of circles. 

The following is the table used : 

7 



98 



MENTAL ARITHMETIC. 



TABLE. 



60 seconds (") = 1 minute, ('). 
60 minutes = 1 degree, (°). 
30 degrees = 1 sign, S. 



12 signs or I _ 
360 degrees J 



1 circle, C. 



1. How many degrees in the circumference of any circle? 
In I the circumference? In | the circumference? 

2. How many seconds in 3' ? In 2 J' ? 

3. How many minutes in 120"? In 90"? 

4. How many minutes in 2° ? In 2^° ? In 5° ? 

5. How many degrees in 180'? In 150'? In 300'? 

6. How many degrees in J of a circle ? In ^ of a circle ? 



MISCELLANEOUS TABLES. 



Counting. 



TABLE. 



12 things = 1 dozen, doz. 

12 dozen = 1 gross, gro. 

12 gross = 1 great gross, G. gro. 
20 things — 1 score. 



Papers and Books. 

TABLE. 

24 sheets = 1 quire. 
20 quires = 1 ream. 

2 reams = 1 bundle. 

5 bundles = 1 bale. 



In printing books — 




When a sheet 


is 


The book is 


And 1 sheet 


folded into 




called 


makes 


2 leaves 




a folio, 


4 pp. (pages). 


4 leaves 




a quarto, or 4to, 


8 pp. 


8 leaves 




an octavo, or 8vo, 


16 pp. 


12 leaves 




a duodecimo, or 12mo, 


24 pp. 


16 leaves 




a 16mo, 


32 pp. 


18 leaves 




an 18mo, 


36 pp. 



DENOMINATE NUMBERS 



&9" 



1. How many are 5^ dozen ? 3^ gro^?^? ; >\ j , . : V '• r :'\ ' ' 

2. How many are 4 score and 10? 

3. How many screws in 2| gross? 

4. How many sheets of paper in ^ quire ? In 2| quires ? 

5. How many quires of paper in ^ ream ? In 3^ reams ? 

6. How many sheets of paper in a ream ? In ^ ream ? 
In I ream? 

7. How many sheets of paper does it take to make a 12mo 
book of 288 pages ? 



MEASUKEMENT OF LUMBER. 

Lumber is known by various names, thus: 1. Boards; 
2. Dimension Stuff; 3. Foot Stuff; 4. Laths; 5. Shingles. 

1. Boards are sold at 1 inch, or less if specified. They 
are — 

1. Stock boards, 12 in. wide ; 

2. Fencing, 6 in. wide ; 

3. Flooring, varying in width ; 

4. Siding or clapboards. 

2. Dimension Stuff consists of — 

1. Scantling or studding, from 2 inches to 4 inches thick 

and from 3 inches to 4 inches wide. 

2. Joists, 2 inches wide and any width. 

3. Plank, 2 inches thick, wider than 4 inches. 

4. Timber, more than 2 inches thick and wider than 4 

inches. 

3. Foot Stuff, sold by linear or the running foot, consists 
of— 

1. Battens, for covering cracks. 

2. Moulding, for finishing 

4. Laths. — 4 feet long, 1^ inches wide, 50 in a bunch. 



100 MENTAL ARITHMETIC. 

o. ShingJes — 4 inciies wide, 250 to the bunch, sold by the 
thousand. Shingles may vary in width, but every 
4 inches in width is reckoned as one shingle. 

Note. — Most lumber is sold by board measure. A board foot is the 
equivalent of a board one foot square and one inch thick. 



Problems. 



I. How many feet of lumber in a board 10 inches wide 
and 12 feet long? 

Note. — The number of inches in length multiplied by the number 

of inches in width, divided by 144, gives the contents in square feet, 

board measure ; and the number of inches in width multiplied by the 

number of feet in length, divided by 12, gives the contents in board 

measure. 

12 V 10 
Hence, -- ^r — = ^^ ^^et, contents of board. 

How many feet, board measure, in the following? 

2. An 8-in. board, 12 ft. long. 

3. A 10-in. board, 12 ft. long. 

4. A 12-in. board, 16 ft. long. 

5. Two 12-in. boards, 12 ft. long. 

6. Two 12-in. boards, 14 ft. long. 

7. Six 10-in. boards, 12ft. long. 

8. Six 8-in. boards, 16 ft. long. 

9. Twenty pieces 3 in. flooring, 16 ft. long. 
10. Thirty pieces 4 in. siding, 16 ft. long. 

II. A plank 2 X 10, 12 ft. 

Note. — This is usually read 2 by 10, 12. 
SuG.— The contents are ?Xi^Xl? ft., or 20 ft. 

Find the value of — 

12. One plank 2X8, 12 ft. 

13. Three planks 2 X 10, 12 ft. 



DENOMINATE NUMBERS. 



101 



14. Six planks 2 X 9, 16 ft. ^ ' ^ > • ^ 

15. One joist 2X6, 12 ft. 

16. Six joists 2 X 8, 14 ft. 

17. Ten pieces scantling 3 X 4, 12 ft. 

18. Twelve pieces scantling 2 X 4, 16 ft. 

19. Four pieces timber 3 X 8, 12 ft. 

20. Twelve pieces timber 6 X 6, 10 ft. 

21. Ten planks 2 X 9, 12 ft. 

22. Six pieces 14-ft. fencing. 

23. Twenty scantling 3 X 4, 16 ft. 

24. Two pieces timber 6 X 8, 32 ft. 

25. Four planks 1| X 10, 12 ft. 

Note. — Lumber less than 1 inch thick is counted as 1 inch ; from 1|- 
to IJ, as 1 J in. ; from If to 1^, as 1| in. ; from 1| to 2, as 2 in. 

A table like the following is generally used by lumber- 
dealers as a matter of convenience : 

LUMBER TABLE. 



Size. 


12 


14 


16 


18 


20 


22 


24 


26 


28 


30 


32 


34 


36 


38 


40 


2X6 


12 


14 


16 


18 


20 

























2X8 


16 


19 


21 


24 
















3X4 


12 


14 


16 


























4X6 


24 


28 




























2X4 


8 






























2X3 

































1. Let the pupil prove the results given. 

2. Let the pupil complete the table. 

SuG. — Let each of the pupils suggest problems under Measurement 
of Lumber for others to solve, including cost of lumber at a certain 
amount per thousand feet. 



.' .1'02 . . ' \ ; : ' Omental arithmetic. 

•"-'•' — ^ ilEviEW Problems. 

1. What is the cost of digging 5 yd. 2 ft. of ditch, at the 
rate of 20 cents a foot ? 

2. What must I pay for 4 yd. 1 ft. of ribbon, at 30 cents 
a yard ? 

3. What is the cost of 6 square yards of cloth, at 5 cents a 
square foot? 

4. How much must I pay for a piece of oilcloth 4^ ft. wide 
and 10 feet long, at 60 cents a square yard ? 

5. When land is worth $1.20 a square rod, what is the cost 
of 2\ acres? 

6. When marble is worth $3 a cubic foot, how much must 
I pay for a block 2 ft. thick, 3 ft. long, and 2 ft. wide ? 

7. How much will 6 lb. 4 oz. of butter cost, at 30 cents a 
pound? 

8. What is the cost of 5 lb. 6 oz. of butter, at 20 cents a 
pound ? 

9. When coal is worth 25 cents a hundredweight, how 
much will I of a ton cost? 

10. What is the cost of 16 pounds of flour, at the rate of 
$3 per hundredweight? 

11. Find the value of 6 tons 3 hundredweight of hay, at 
$15 a ton. 

12. How much will 5 tons 4 cwt. of coal cost, at $5 a 
ton? 

13. What is the value of 2 ounces 2 pennyweights of gold, 
at $1 per pennyweight ? 

14. How much will 2 quarts and 1 pint of molasses cost, 
at 18 cents a quart ? 

15. How much will 1 gallon and 1 quart of oil cost, at 8 
cents a quart? 

16. Find the cost of 2 gallons 3 quarts of wine, at $1.20 a 
gallon. 



DENOMINATE NUMBERS. 103 

17. If I buy chestnuts at $2 a bushel and sell them at 10 
cents a quart, how much do I make on a bushel ? 

18. How much do I make on 3 bushels 2 pecks of chest- 
nuts that cost me $2.50 a bushel, if I sell them at 12 cents a 
quart ? 

19. What is the cost of 5 bu. 3 pk. of salt, at 2 cents a 
quart ? 

20. How many half-pint bottles can be filled from 3 gal- 
lons of water ? 

21. What is the cost of 8 buttons, at 15 cents a dozen? 

22. I bought collars at 25 cents a box, each box contain- 
ing 10 collars ; how much per box will I gain by selling the 
collars at 4 cents apiece ? 

23. If I buy 3 shovels at the rate of $8 a dozen, how much 
do they cost me ? 

24. How many feet high is a horse that is 16 hands high ? 

25. When apples sell at 15 cents a peck, how much will 
2 bushels cost ? 

26. What is the cost of 5 hats, at the rate of $24 a dozen ? 

27. A stationer buys lead-pencils at $4.50 a gross and re- 
tails them at 5 cents each ; what is his profit on a gross? 

28. A stationer buys lead-pencils at 50 cents a dozen and 
sells them at 5 cents apiece ; how much does he make on 3 
dozen ? 

29. If a merchant buys paper at $1.20 a ream and sells it 
at 10 cents a quire, what is his profit on a quire ? 

30. A merchant buys paper at $1.40 a ream and sells it at 
a cent a sheet ; what is his profit on 1^ reams ? 

31. How many hands high is a horse that is 5 feet 3 inches 
high? 

32. A grocer bought cranberries at $4.80 a bushel ; how 
must he sell them to gain 3 cents a quart? 



104 MENTAL ARITHMETIC. 

LONGITUDE AND TIME. 

Longitude is the distance east or west from a given 
meridian. 

A meridian is a line passing from pole to pole through the 
equator. 

Since the earth revolves on its axis from west to east in 24 
hours, the sun appears to pass around the earth, or through 
360 degrees of longitude, in the same time, or through 15 
degrees of longitude in one hour of time. In 1 minute of 
time it passes through ^j^ of 15°, or J of a degree, equal to 
15' of longitude; and in 1 second of time it passes through 
15" of longitude ; hence — 

For a difference of There is a difference of 

15° in longitude, 1 hour of time. 

15' *•' " 1 minute " " 

15" ** " 1 second " " 

Or, 
1° of longitude is equivalent to 4 minutes in time. 

1' of longitude " " " 4 seconds in time. 

1" of longitude " " " -^^ of a second in time. 

1. Over how many degrees of the earth's surface does the 
sun pass in 24 hours? In 12 hr. ? In 6 hr. ? 

2. Over how many degrees does the sun pass in 1 hr. ? In 
4 hr. ? In 10 hr. ? 

3. Over how many degrees does the sun pass in 2 hr. ? In 
\ hr. ? In 20 min. ? In 10 min. ? 

4. How many degrees of longitude are equivalent to 1 hr. ? 
2hr.? 4hr.? 10 hr.? 8 hr. ? 

5. When it is noon at any place, what time is it 15° east? 
15° west? 30° west? 30° east? 

6. What is the difference in time between two places which 
have a difference of 15° longitude ? 30° ? 60° ? 120° ? 



PERCENTAGE. 105 

7. When it is 4 o'clock at St. Louis, what o'clock is it 30°. 
east? 30° west? 60° west? 

8. When it is 6 o'clock at St. Paul, how far east or west is 
it 4 o'clock ? 3 o'clock ? 7 o'clock ? 9 o'clock ? 

9. What difference in longitude indicates a difference of 4 
minutes in time ? 1 minute? 8 minutes? 1 hour? ^hour? 

10. How far east or w^est of Indianapolis is a man whose 
watch is 1^ hours slow, if he has correct Indianapolis time? 

11. What is the difference in time of sunrise between two 
places 36° of longitude apart ? 

12. If after traveling a month I find my watch 2^ hours 
faster than local time, am I east or west of the starting-point, 
and how many degrees ? 

13. Two men look at their watches at the same moment ; 
both have correct time: one's watch marks 9 o'clock 15 min-. 
utes ; the other's marks 11 o'clock 30 minutes ; how many 
degrees apart are they, and which is the farther west ? 



PBRCBNTAGB. 



Percentage is the name applied to computations in which 
the unit of measure is 100. 

Per cent, is an abbreviation of the Latin phrase jper cen- 
turrif meaning by the hundred. 

Note. — One per cent, of any number is jj^ of it ; 5 per cent, ia 
xf^, and so on. 

1. How many hundredths of a number is 2 per cent, of it? 
10 per cent. ? 20 per cent.? 40 per cent. ? 

2. How many hundredths of anything is 3 per cent, of it? 
12A per cent. ? 50 per cent. ? 150 per cent. ? J of 1 per 
cent. ? 



106 MENTAL ARITHMETIC. 

3. What per cent, of a number is j^^ of it? j^ij- of it? 

191 

j-|ofit? ^Voofit? 

Per cent, is usually expressed % ; thus 12% is read 12 per cent. 

4. How many hundredths in 25% ? In 50% ? In 6^% ? 
In 331%? In56i%? 

5. What per cent, of a number is ^ of it ? 
Suggestion.— J = xVa> or 25 % . 

6. What per cent, of a number is ^ of it? |? y^^? ^V^ 

3V? V V V t-J u? 

7. What fractional part of a number is 30% of it? 
Suggestion. — 30% — ^^j^ = j\ ; hence 30% of a number is ^5 of it. 

8. What fractional part of a number is 10% of it ? 20% ? 
5%? 25%? 75%? 60%? 37i%? 33^%? 12^%? 
6^%? 87^%? 66f%? 140%? 

9. What part of a number is I % of it ? 
Suggestion.— J % =iX tU = jU- 

10. What part of a number is i% of it? |%? i%? 
|%? |%? 



1. 

To Find the Percentage. 

1. What is 30% of 40 barrels of apples? 

Analysis. — 30% is j\%, or y%; j\ of 40 barrels of apples is 12 
barrels. 

What is— 

2. 10% of 90? 7. 15% of 200? 

3. 5% of 40? 8. 8% of 75? 

4. 6% of 50? 9. 12^% of 48? 

5. 20% of 30? 10. 33|% of 120? 

6. 25% of 84? 11. 55% of 400? 



PERCENTAGE. 107 

12. What is 62|% of 16 sheep ? 

13. What is 6% of $350? 

14. What is 5% of $140? 

15. Henry had $120, but speut 50% of it for a watch; 
what did the watch cost ? 

16. A farmer sold a cow worth $40 at a loss of 10% ; what 
did he receive for her ? 

17. A merchant bought goods at 50 cents a yard and sold 
it at a gain of 20 per cent. ; what price did he receive for the 
goods ? 

18. A farmer who had 500 bushels of wheat sold 24 per 
cent, of it ; how much did he sell ? 

19. A manufacturer who paid his laborers $1.50 a day 
advanced their wages 10 per cent. ; how much did he pay 
them then ? 

20. If laborers who receive $2 a day have their wages re- 
duced 12 per cent., how much do they then receive? 

21. A piece of cloth measuring 150 yards shrunk 5 per 
cent. ; how long was it then ? 

22. I buy a bill of goods for $52, but get a discount of 5 
per cent, for paying cash ; what do I pay to cancel the bill ? 

23. Harry had 40 chickens, but they increased 150 per 
cent. ; how many had he then ? 

24. Having $300, 1 paid 30% for board, 20% for clothing, 
and 15% for other expenses; how many dollars have I re- 
maining? 

25. Oliver bought silk for 60 cents and sold it at a gain of 
20% ; what did he get for it? 

26. If I buy 6 barrels of flour for $30 and sell it at an 
advance of 25%, how much do I get per barrel? 

27. A collector receives 5% of a bill of $4820 ; how much 
money does he receive ? 

28. A man's wages were $2 a day : they were reduced 10%, 
and then increased 10% ; what are his wages now? 



108 MENTAL ARITHMETIC. 

2. 

To Find the Kate. 

1. What per cent, of 45 is 9? - 

Analysis.— 9 is | of 45, or } of 100% of 45, which is 20% of 45. 

What per cent. — 

2. Of 30 is 6? 8. Of $25 are $3? 

3. Of 40 is 10? 9. Of 60 sheep are 5 sheep? 

4. Of 50 is 5 ? 10. Of 65 books are 13 books? 

5. Of 24 A. are 18 A.? 11. Of 200 is 50? 

6. Of 180 lb. are 60 lb. ? 12. Of 2| is ^ ? 

7. Of 48 ft. are 6 ft. ? 13. Of 37| cents are 10 centa? 

14. What per cent, of $2 are 60 cents ? 

15. What per cent, of 5 quarts are 3 pints? 

16. What per cent, of money is | of it? 

17. A boy carried 5 dozen of eggs to a store, but in falling 
he broke 6 eggs ; what per cent, did he break? 

18. I had 80 sheep, but I have sold 16 ; what per cent, did 
I sell? 

19. What per cent, of $3| is | of a dollar? 

20. If I have $50 and spend $10 for books and $6 for 
board, what per cent, of my money remains? 

21. If from a barrel containing 30 gallons I take 12 gal- 
lons, what per cent, do I take? 

22. f of 80 is what per cent, of ^ of 120? 

23. I of 24 is what per cent, of | of 60? 

24. I of 40 is what per cent, of | of 180? 

25. A man who had earned $520 spent at one time 25% 
and at another 30% of it; how much did he spend each 
time? 

26. A man whose income was $1600 a year spent $300 for 
board, $160 for books, and $40 for travel ; what per cent, did 
he spend for each, and what per cent, did he save ? 



PERCENTAGE. 109 

3. 

To Find the Base. 

I. 42 is 14% of what number? 

Analysis 1.-14% is ^q%j or ^V- If 42 is ^^ of a number, -j^ of 
that number is ^ of 42, or 6, and f §, or the number, is 50 times 6, or 300. 

Analysis 2.— If 14%, or yV(J> of a number is 42, yj^ is x^^ of 42, or 
3, and J^§, or the number, is 100 times 3, or 300. 

Note. — Either analysis may be used. 

2. 16 is 4% of what? 6. 20 bu. are 33|% of what? 

3. 25 is 5% of what? 7. 8 qt. are 40% of what? 

4. 40 is 32% of what? 8. 6^ is 13% of what? 

5. 32 lbs. are 8% of what? 9. 7^ is 12% of what? 
10. 10 quarts are 40% of how many gallons? 

II. A farmer gained $6 on a horse, which was 3% of his 
value; what was the value? 

12. $18 is 4 per cent, of my money ; how much have I? 

13. I rent a house at $120, which is 8% of the cost of the 
house ; how much did it cost ? 

14. George spends $40 a year for cigars, which is 8% of 
his income ; what is his income ? 

15. I paid $30 for a chain, which is 24% of the cost of my 
watch ; what is the cost of my watch ? 

16. My board costs me $310 a year, which is 40% of my 
salary ; what is my salary ? 

17. A farmer sold 160 bushels of corn, which was 32% of 
his crop ; what was the crop ? 

18. If by selling cloth at 20% gain I make 14 cents, what 
was the cost of the cloth ? 

19. Mr. Henry, owning ^ of a ship, sold 30% of his share 
for $9000 ; what was the value of the whole ship ? 

20. 2 pk. 3 qt. was 25% of the nuts Harry gathered; how 
many did he gather ? 



110 MENTAL ARITHMETIC. 

Review Problems. 

1. A miller sold flour at $6, and thereby gained 20% ; what 
was the cost of the flour? 

Solution.— 20% equals J ; |, the cost, plus J, the gain, equals f , or 
the selling price. If ^ of the cost of the flour is $6, | is J of $6, or 
$1, and I, or the cost, is 5 times $1, or $5. 

2. A horse was sold at $40 more than cost, which was at a 
gain of 50% ; what was the cost? 

3. A knife was sold at 30 cents less than cost, which was 
at a loss of 60% ; what was the cost? 

4. A miller sold some damaged flour at $6, and thereby 
lost 25% ; what was the price of the flour? 

5. A bookseller retails books at 75 cents each, and thereby 
gains 25% on cost; what did they cost him? 

6. I bought a cow of my neighbor and sold her for $46, 
which was at a gain of 15%; how much^did I pay for 
her? 

7. If a merchant sells damaged cloth at $2 a yard, and 
thereby loses 20%, what was the cost of the cloth? 

8. If a merchant make 8% by selling sheeting at 27 cents 
a yard, what did he pay for it? 

9. What number, increased by 20% of itself, equals 48? 

10. What number, increased by 75% of itself, equals 28? 

11. 80 is 25% more than what number? 

12. What number, diminished by 30% of itself, equals 28? 

13. What number, diminished by 75% of itself, equals 16? 

14. 63 is 30% less than what number? 

15. What number, increased by the difference between 
70% and 40% of itself, equals 52? 

16. When cloth is sold at $1.65 a yard, at a gain of 10%, 
what is the cost ? 

17. I sold flour at $7.50 a barrel, at a gain of 20% ; what 
was the cost of the flour ? 



PROFIT AKD LOSS. Ill 

18. Two boys have 110 marbles: one has 20% more than 
the other ; how many has each ? 

19. What is the length of a ditch whose length, increased 
by 40% and 5 feet, equals 89 feet? 

20. Ahorse and carriage cost $188: the horse cost 35% 
more than the carriage ; how much did each cost ? 

21. In a school of 115 pupils the number of boys is 30% 
greater than the number of girls ; how many are there of 
each ? 

22. What fraction, increased by 33^ % of itself, equals 4 ? 

23. Kudolph has 36 marbles, which is 25% less than Wil- 
liam's number ; how many has William ? 



PROFIT AND LOSS. 

1. I bought corn at 40 cents a bushel and sold it at a gain 
of 12^% ; how much did I get for it? 

101 

Analysis 1. — If I gain 12|%, I gain —^, or |. |, the cost, plus J, 

the gain, equals |, or the selling price, f of 40 cents is 45 cents. 

Analysis 2. — 12|% equals J ; | of 40 cents is 5 cents ; hence I sold 
the corn for 40 cents plus 5 cents, or 45 cents. 

2. If I buy a cow for $40, how must I sell her to gain 
20%? 

3. How must I sell eggs for which I pay 20 cents a dozen, 
so as to gain 30% ? 

4. If a boy pay 30 cents a dozen for oranges, how must he 
sell them apiece so as to gain 60 % ? 

5. If I buy 10* barrels of apples for $25, how must I sell 
them per barrel to gain 20% ? 

6. A merchant paid 80 cents a yard for silk ; how must he 
sell it to gain 12^% ? 



112 MENTAL ARITHMETIC. 

7. A house cost $2000, and was sold at a gain of 5\% ; 
at what price was it sold? 



1. If a sheep is bought for $4 and sold for $6, what is the 
gain per cent. ? 

Analysis. — If a sheep is bought for $4 and sold for $6, the gain is 
16 minus $4, or $2. $2 is J of $4, or J of 100%, which is 50%. 

2. If I buy a cow at $30 and sell her at $36, what is the 
gain per cent. ? 

3. When a dealer buys melons at $12 a hundred and sells 
them at $16 a hundred, what is his gain per cent.? 

4. What is the gain per cent, on a book that is bought at 
40 cents and sold at 50 cents ? 

^5. A produce dealer buys eggs at 25 cents a dozen and 
sells them at 30 cents a dozen ; what is his gain per cent. ? 

6. A merchant buys potatoes at 60 cents a bushel and re- 
tails them at 25 cents a peck ; what is his gain per cent. ? 

7. A butcher buys beef at 6 cents a pound and retails it at 
10 cents a pound ; what is his gain per cent. ? 

8. A huckster buys oranges at 25 cents a dozen and retails 
them at 5 cents apiece ; what is his gain per cent.? 

9. When a bushel of chestnuts costs $2 and is retailed at 
10 cents a quart, what is the gain per cent. ? 

10. I sold I of a farm for what the whole farm cost me ; 
what was the gain per cent. ? 

11. What is the loss per cent, when | of an article is sold 
for the value of | of it ? 

12. A piano was bought for $250, and after several years' 
use was sold for $150 ; what was the loss per cent. ? 

13. A dealer bought flour at $6 a barrel and sold it at 90 
cents a sack (| of a barrel) ; what was his gain per cent. ? 

14. How must goods costing $3 be marked so that the seller 
may drop 10% in his price and still make 20% ? 



PROFIT AND LOSS. 113 

1. A boy sold a book for 15 cents more than cost, and 
gained 10% ; what was the cost? 

Analysis. — 10%, or the gain, is -f^ of the cost. If -^q of the cost is 
15 cents, {^, or the cost, is 10 times 15 cents, or $1.50. 

2. If a farmer sell a horse at $30 more than cost, and 
thereby gain 15%, what was the cost? 

3. A miller sold damaged flour at $1.50 less than its actual 
Talue, and thereby lost 25% ; what was the real value of the 
flour? 

4. A piano was sold at $30 less than cost, which was at a 
loss of 15% ; what was the cost? 

5. If I lose 40 cents by selling an article at 12^% below 
its value, what is the value ? 

6. A man's property having increased 30% in value, he 
sold it for $600 more than cost ; what did it cost him ? 

7. Property having depreciated in value, I sold my house 
for $2500 less than cost, and thereby lost 50% ; what was the 
cost of the house ? 



1. If by selling eggs at 30 cents a dozen I make 20%, 
what did the eggs cost me? 

Analysis. — Since I gained 20%, or | of the cost, I sold the eggs at 
I + -J, or f , of the cost. If f of the cost is 30 cents, ^ of the cost is J 
of 30 cents, or 5 cents, and f , or the cost, is 5 times 5 cents, or 25 
cents. 

2. A jeweler sold a watch for $75, thereby gaining 25% ; 
what did he pay for the watch? 

3. If by selling coffee at 28 cents I make 12%, what was 
the cost of the coffee? 

4. A boy sold his knife for 40 cents, which was 20% less 
than cost ; at what price must he have sold it to gain 10% ? 

5. A gentleman sold a horse for $168 and a carriage for 
$84; he gained on each 12% ; what was the cost of each? 

8 * 



114 MENTAL ARITHMETIC. 

6. If by selling two articles at $10 each I gain on the first 
25% and on the second lose 25%, do I gain or lose? 

7. By selling sugar at 7 cents a pound I gain 16|% ; what 
is the cost of the sugar ? 

8. An investment pays a profit of ^200 at 8% ; what would 
it pay at 9J%? 

9. A miller had 50 barrels of flour worth $6 a barrel : 10 
barrels were destroyed by fire ; how must he sell the re- 
mainder to clear 8% on the lot? 

10. A gentleman bought a watch for $56, which was 20% 
less than its value; how must he sell it to gain 10% on its 
value ? 



COMMISSION AND BROKEKAGE. 

1. An attorney collects a debt of $400 on a commission of 
3% ; what is his commission? 

SuG.— His commission is 3% of $400. 

2. A tax-collector receives a commission of 2% for collect- 
ing $8000 taxes ; what is his commission ? 

3. If an agent sell a house for $5000 and charge a 2^ % 
commission, what amount does he receive ? 

4. What is the expense of collecting a debt of $1200 on a 
commission of 5% ? 

5. A commission merchant sells $650 worth of berries on 
10% commission ; what is his commission? 



1. An agent charges $12 for collecting a bill of $240 ; what 
per cent, does he charge ? 

Analysis. — If he charges $12 for collecting $240, on$l he charges 
aio oi $12, or -i^^, equal to ^^y or 5%. 

2. A lawyer collects a debt of $600 for $24 ; what per cent, 
does he charge ? 



COMMISSION AND BROKERAGE. 115 

3. If a tax-collector receives $54 for collecting $1800, what 
is his rate of commission ? 

4. A commission merchant charges $40 for selling $800 
worth of produce ; what is his rate of commission ? 

5. If an agent charge $25 for selling a property worth 
$1000, what is his rate of commission? 



1. A tax-collector receives $180 for collecting taxes on a 
3 % commission ; what is the amount collected ? 

Analysis. — 3 % --= j|^ ; if $180 is jf^ of the amount collected, j-J^ 
is i of $180, or $60, and j§§ is 100 times $60, or $6000. 

2. A commission merchant charges $42 for selling goods on 
10% commission; what amount did he sell? 

3. An architect charges $40 on a 2% commission for the 
plans of a building; what is the cost of the building? 

4. An agent sells goods on a 4% commission; his salary 
for the first month is $120 ; what amount of goods did he 
sell? 

5. An attorney charges his client $60 on a 5% commission 
for collecting a debt ; what is the amount of the debt ? 



1. What amount of goods can I purchase for $520, if my 
agent charges me 4% on the amount expended? 

Analysis. — 4%, or -}^, of the amount expended, plus |f, equals ff, 
or $520 ; 2V of the amount expended equals ^V of $520, or $20, and ff , 
or the amount expended, equals 25 times $20, or $500. 

2. What amount of goods can be purchased for $420, if the 
agent's commission is 5%? 

3. If an agent charge me 2% for buying goods, how much 
will he purchase for $510 ? 

4. I send an agent $315 to purchase goods for me : he 
charges 5% commission; what amount of goods does he 
purchase ? 



116 MENTAL ARITHMETIC. 



STOCKS. 



Note 1. — When stock is at a premium its market value is the par 
value plus the premium ; and when at a discount, the market value is 
the par value less the discount. 

Note 2. — When the par value is not mentioned it is understood to 
be $100 a share. 

1. When stock is at a premium of 10%, what is the value 
of $1 worth? $100? $50? 

2. When stock is at a discount of 20%, what is $1 worth? 
$100? $10? $50? 

3. What is the cost of 10 shares of railway stock at 15% 
premium ? 

4. What is the cost of 5 shares of railway stock at 20% 
premium ? 

5. What is the cost of 10 shares of stock at 10% discount ? 



1. If I buy stock at 92 and sell at 96, and thereby make 
$120, how many shares do I buy ? 

Analysis. — If I buy stock at 92 and sell it at 96, I gain on each 
share $96 — $92, or $4 ; I therefore buy as many shares as $4 is con- 
tained times in $120, or 30 shares. 

2. A man bought stock at 90 and sold at 85, thereby losing 
$60 ; how many shares had he ? 

3. A speculator bought bank stock at 120 and sold at 127, 
thereby gaining $840 ; how many shares had he ? 

4. If Mr. A buys stock at 92 and is compelled to sell at 88, 
thereby losing $200, how many shares has he ? 



1. A company declares a dividend of 6% ; how much div- 
idend does Mr. AValters get, if he owns 30 shares ? 
* SuG. — Since the dividend is |6 on a share, Mr. Walters gets 30 times 
$6, or $180. 



STOCKS. 117 

2. If a bank declare a dividend of 5%, what is that man's 
dividend who has 20 shares of stock ? 

3. A railway company declares a dividend of 3 per cent. ; 
what is Mr. A's dividend, if he own 80 shares of stock ? 

4. A manufacturing company declares a dividend of 8 per 
cent. ; what is my dividend, if I hold 40 shares of stock ? 



1. A company declares a dividend of 4% ; how much stock 
do I own, if I receive S400 dividend ? 

Analysis. — Since mj dividend is 4 % , or -}^, $400 is -f^ of my stock ; 
If, or the value of my stock, is 25 times |400, or |10,000. 

2. If Mr. B's share of a 5% dividend is $200, how much 
stock does he ovfw ? 

3. A stockholder in a bank receives $60 as his share of a 
3% dividend ; how much stock does he own ? 

4. Mr. Lindsey receives $540 as his share of a 5% divi- 
dend ; how much stock does he own ? 



1. If $500 worth of stock earn $30, what rate of dividend 
may be declared ? 

Analysis.— If $500 gain $30, $1 gains ^J^ of $30, or ^^^, equal to 
Tf(T,or6%. 

2. When $800 earns a profit of $40, what rate of dividend 
may be declared ? 

3. What rate of dividend may be declared on stock, if 
$5000 of it gain $400 ? 

4. 10 shares of bank stock, worth $100 each, pay an an- 
nual dividend of $80 ; what is the rate per cent. ? 



1. What amount of stock at 10% premium can I buy for 
$440? 

Analysis. — At 10% premium, stock is worth $110 a share; hence 
$440 will buy as many shares as $110 is contained times in $440, or 4 
shares. 



118 MENTAL ARITHMETIC. 

2. When stock is selling at 30% premium, what amount 
may be bought for $6500 ? 

3. If railroad stock sells at 25% discount, how much may 
be bought for $1500? 

4. How much stock at 20% premium may be bought for 
$2400, if the shares are $50 each ? 



INSURANCE. 

1. How much will it cost to insure a house worth $3000 
atl|%? 

SuG.— It will cost li% of $3000, or $37.50. 

2. A house was insured for $4000 at 1% ; what was the 
premium, or cost of insuring ? 

3. What is the premium on a house worth $4500, insured 
for 5 years at 1A% ? 

4. If I pay $30 insurance on a $3000 house, what is the 
rate? 

5. A merchant insured his store for $4000, and paid $100 
insurance ; what was the rate ? 

6. If I pay $75 insurance on a property worth $5000, what 
is the rate ? 

7. Mr. Jenkins insured his furniture at 2% and paid a 
premium of $20 ; what is the amount of the insurance ? 

8. If a man pay $30 insurance at 1^%, what amount of 
insurance does he get? 

9. The insurance on a barn at | % costs $18 ; what is the 
amount of insurance ? 



INTEREST. 



Interest is a certain percentage paid for the use of money. 
It is estimated at a certain rate per cent, for each year. 
The Principal is the sum on which the interest is paid. 



INTEREST. 119 

The Rate is the percentage paid per annum. 
The Amount is the sum of the principal and the interest. 
In computing interest 30 days are considered a month, and 
12 months a year. 

General Method. 

1. What is the yearly interest of $40 at 6% ? 

Analysis.— At 6%, yf^, or -j%, of the principal equals the interest. 
/o of 140 is $2.40 ; hence the interest of |40 for 1 year at 6^ is $2.40. 

2. What is the yearly interest of $300 at 6% ? 

• 3. What is the yearly interest of $225 at 4% ? 

4. What is the yearly interest of $50 at 8% ? 

5. What is the yearly interest of $60 at 5% ? 

6. What is the yearly interest of $40 at 7% ? 

7. What is the yearly interest of $120 at 12^% ? 

8. What is the yearly interest of $500 at 5^% ? 

9. What is the yearly interest of $140 at 3% ? 
10. What is the yearly interest of $90.50 at 6% ? 



1. What is the interest of $500 for 3 yr. at 6% ? 

Solution. — At 6%, yf^y, or /q, of the principal equals the interest 
for 1 year ; and for 3 years, 3 times /^, or /q, of the principal equals the 
interest ; ^^ of |500 is $90 ; hence the interest of $500 for 3 yr. at 6% 
is $90. 

2. What is the interest of $40 for 2 yr. at 5% ? 

3. What is the interest of $100 for 3 yr. at 6% ? 

4. What is the interest of $80 for 5 yr. at 4% ? 

5. What is the interest of $90 for 4 yr. at 5% ? 

6. What is the interest of $120 for 3 yr. at 5% ? 

7. What is the interest of $350 for 2 yr. at 8% ? 

8. What is the interest of $200 for 2^ yr. at 6% ? 

9. What is the interest of $400 for 31 yr. at 6% ? 



120 MENTAL ARITHMETIC. 

10. What is the interest of $300 for 3| yr. at 5% ? 

11. What is the interest of $300 for 3 yr. 4 mo. at 6% ? 

Solution. — 2 yr. 4 mo. equal 2 J yr. If the interest for one year is 
6%, for 2J yr. it is 2} times 0%, or 14% ; 14% of $300 is $42. There- 
fore, etc. 

What is the interest — 

12. Of $200 for 2 yr. 8 mo. at 6% ? 

13. Of $250 for 3 yr. 6 mo. at 4% ? 

14. Of $400 for 4 yr. 3 mo. at 5% ? 

15. Of $120 for 6 yr. 3 mo. at 8% ? 

16. Of $1000 for 3 yr. 7 mo. at 6% ? 

17. Of $450 for 2 yr. 5 mo. at 8% ? 

18. Of $500 for 2 yr. 9 mo. at 5% ? 

19. Of $20 for 1 yr. 6 mo. at 7% ? 

20. Of $340 for 2 yr. 6 mo. at 6% ? 

21. What is the interest of $300 for 2 yr. 3 mo. 18 days 
at 5% ? 

Solution. — 18 days equal |§, or |, of a month ; hence 3 months and 
18 days equal 3f months, or -^/ months. J of a month equals ^ of j^tj 
of a year, or -^j^ of a year, and -y- mouths equal 18 times ^V> ^^ t%> ^^ 
a year ; hence 2 yr. 3 mo. and 18 days equal 2j% years. If the in- 
terest for 1 year is 5%, for 2^^^ years it is 2^ times 5%, or 12^% ; 
12|% of $300 is $37.50. Therefore, etc. 

What is the interest — 

22. Of $400 for 5 yr. 3 mo. 18 da. at 10% ? 

23. Of $500 for 3 yr. 7 mo. 6 da. at 5% ? 

24. Of $600 for 1 yr. 6 mo. 20 da. at 9% ? 

25. Of $300 for 2 yr. 7 mo. 6 da. at 5% ? 

26. Of $150 for 4 yr. 3 mo. 20 da. at 6% ? 

27. What is the amount of $250 for 6 yr. at 5% ? 

Note. — The amount is the sum of the principal and the interest. 

Solution.— For 6 yr. at 5%, 30%, or j%, of the principal equals the 
interest. ^^ of $250 is $75 ; $250 plus $75 are $325, the amount. 



INTEREST. 121 

What is the amount — 

28. Of $240 for 4 yr. at 10% ? 

29. Of $650 for 5 yr. at 6% ? 
.30. Of $800 for 6^ yr. at 4% ? 

31. Of $600 for 3 yr. 6 mo. at 6% ? 

32. Of $250 for 4 yr. 3 mo. at 5% ? 

33. Of $600 for 8 yr. 5 mo. at 6% ? 

34. Of $1000 for 3 yr. 2 mo. 12 da. at 5% ? 

35. Of $800 for 4 yr. 24 da. at 5% ? 

36. Of $200 for 2 yr. 7 mo. 6 da. at 5% ? 

37. A has $400, which amount is f of C's money ; what is 
the interest on each one's money for 3^ yr. at 6% ? 

38. I build a house for $4000 when money is worth 5% ; 
what is the yearly interest on the value of the house ? 

39. If a house cost me $3000 when money is worth 6%, 
and my taxes and insurance amount to 3%, what does the 
use of the house cost me each year ? 



Interest by Days. 

When the number of days for which money draws interest 
does not exceed 123, a short method much used is that of 
moving the decimal point two places to the left in the princi- 
pal to find the interest for 60 days at 6%. 

Thus, the interest of $200 for 60 days at 6% is $2.00. For 
63 days it is $2.00 and -^^ of $2.00, or $2.10, because the 3 
days is ^'^ of 60 days. 

Find the interest of — 

1. $360 for 60 da. at 6%. 6. $200 for 93 da. at 6%. 

2. $720 for 60 da. at 6%. 7. $400 for 60 da. at 5%. 

3. $407 for 60 da. at 6%. 8. $600 for 60 da. at 10%. 

4. $100 for 63 da. at 6%. 9. $800 for 63 da. at 5%. 

5. $200 for 33 da. at 6%. 10. $200 for 93 da. at 5%. 



122 MENTAL ARITHMETIC. 

11. $600 for 93 da. at 8%. 18. $660 for 93 da. at 6%. 

12. $100 for 63 da. at 8%. 19. $330 for 93 da. at 8%. 

13. $200 for 33 da. at 8%. 20. $550 for 60 da. at 6%. 

14. $600 for 60 da. at 7%. 21. $600 for 45 da. at 6%. 

15. $900 for 63 da. at 7%. 22. $800 for 63 da. at 6%. 

16. $1200 for 95 da. at 6%. 23. $500 for 63 da. at 5%. 

17. $840 for 63 da. at 6%. 24. $400 for 33 da. at 10%. 



GENERAL PROBLEMS IN INTEREST. 

1, 

To Find the Rate. 

I. At what per cent, will $200 gain $36 in 3 years? 

Solution 1.— The interest on $200 for 1 yr. at 1 % is |2, and for 3 
yr. it is 3 times $2, or $6. For $200 to gain $36 will therefore require 
a rate of as many per cent, as $6 is contained times in $36, or 6%. 

Solution 2. — At 1 % the interest on $1 for 3 years is 3 cents, and 
on $200 it is 200 times 3 cents, or $6 ; it will therefore require as many 
per cent, as $6 is contained times in $36, or 6 % . 

2. At what per cent, will $300 gain $30 in 2 yr. ? 

3. At what per cent, will $150 gain $27 in 3 yr.? 

4. At what per cent, will $180 gain $36 in 6 yr.? 

5. At what per cent, will $40 gain $16 in 8 yr.? 

6. At what per cent, will $250 gain $35 in 3| yr. ? 

7. At what per cent, will $60 gain $12 in 5 yr. ? 

8. At what per cent, will $80 gain $10 in 1\ yr. ? 

9. At what per cent, will $90 amount to $108 in 4 yr.? 
10. At what per cent, will $500 amount to $605 in 3^ yr. ? 

II. At what per cent, will $30 in 4 yr. 2 mo. amount to 
$37.50? . 

12. At what per cent, will $300 in 5 yr. 3 mo. amount to 
$363? 



INTEREST. 123 

13. At what per cent, will a principal double itself in 10 
yr. ? 20 yr. ? 4 yr. ? 5 yr. ? 25 yr. ? 

Note. — In order to double itself it must gain 100^, or once itself. 

14. At what per cent, will a principal treble itself in 4 yr. ? 
5yr.? 10 yr.? 20 yr.? 25 yr.? 40 yr.? 

15. At what per cent, will a principal quadruple itself in 
4yr.? 5yr.? 10 yr.? 15 yr.? 30 yr.? 

16. At what rate will the interest of $600 for 3 yr. 3 mo. 
6 days be $102? 

17. The amount of a certain principal for 5 years at a cer- 
tain per cent, is $750, and for 3 years, $690 ; what are the 
principal and the rate per cent.? 

18. The amount of a certain principal for 4 years at a cer- 
tain per cent, is $650, and for 7 years, $710; what are the 
principal and the rate per cent. ? 



2. 

To Find the Time. 



I. In what time will $300 at 6% give $54 interest? 

Solution. — $300 in 1 year, at 6^, will yield $18 interest, hence to 
yield $54 will require as many years as 18 is contained times in 54, 
or 3 years. 

2. In what time will $100 at 5% yield $15 interest? 

3. In what time will $200 at 6% yield $24 interest? 

4. In what time will $100 at 4% yield $32 interest ? 

5. In what time will $300 at 5% yield $60 interest? 

6. In what time will $140 at 5% yield $28 interest? 

7. In what time will $150 at 6% yield $45 interest? 

8. In what time will $60 at 5% yield $7.50 interest? 

9. In what time will $25 at 6% yield $7.50 interest? 
10. In what time will $200 at 7% yield $42 interest? 

II. In what time will $300 at 6% yield $24 interest? 



124 MENTAL ARITHMETIC. 

12. In what time will $400 at 6% amount to $460? 

13. In what time will $540 at 5% amount to $621 ? 

14. In what time will any principal double itself at 5% ? 
10%? 20%? 40%? 25%? 50%? 

15. In what time will any principal treble itself at 5%? 
10%? 20%? 25%? 40%? 16|%? 

16. In what time will a principal quadruple itself at 5% ? 
10%? 15%? 20%? 25%? 12^%? 

17. The amount of a certain principal for a certain time 
at 5% is $600, and for the same time at 8%, $660 ; what are 
the principal and the time ? 

18. If I invest my money in bank stock yielding a divi- 
dend of 10 per cent., it will in a certain time amount to 
$9000; but if I loan it at 6% for the same time, it will 
amount to only $7800 ; what is the amount of money, and 
what the time? 



3. 

To Find the Principal. 

1. What principal will in 6 yr. 6 mo. at 4% give $52 
interest ? 

Solution 1. — 6 yr. 6 mo. equals 6^ years. The interest on $1 for 
6 1 yr. at 4% is 26 cents ; hence $52 is the interest on as many dollars 
for the given time as 26 cents is contained times in $52, or $200. 

Solution 2. — For 6 yr. 6 mo., or 6^ yr., at 4%, ^f of the principal 
equals the interest. Since $52 is || of the principal, -^q of the principal 
is yV of $52, or $4, and |^, or the principal, is 50 times $4, or $200. 

2. What principal will in 5 yr. at 4% yield $20 interest? 
$60? $200? $1200? 

3. What principal will in 6 yr. at 5% yield $60 interest? 
$150? $600? $9? 

4. What principal will in 8 yr. at 4% yield $80 interest? 
$160? $640? $200? 



PRESENT WORTH. 126 

5. What principal will in 6 yr. 8 mo. at 6% yield $100 
interest? $120? $640? $1200? 

6. A man pays annually on borrowed money $450, at 5% ; 
how much has he borrowed ? 

7. How much money has a man on interest at 5%, if his 
annual income is $1030? 

8. A gentleman sold his farm, to be paid for at the end of 
1 yr. 4 mo., interest at 6%. When payment was made the 
amount was $12,960 ; what was the price of the farm ? 

9. How much money must be put on interest to amount to 
$690 in 3 years at 5% ? 

10. What principal will in 1 yr. 6 mo. at 8% amount to 
$224? 

11. A debt which has been drawing interest for 6 yr. 4 mo. 
at 6% amounts to $552 ; what is the debt? 

12. Mr. Williams invested | of his money in the lumber 
trade, and at the end of 2 yr. 6 mo. sold his share for $10,800, 
at a profit of 8 per cent, yearly gain ; how much money had 
he at first ? 



PRESENT WORTH. 



The Present Worth of a debt is a sum which, if placed on 
interest at the legal rate for the given time, will amount to 
the face of the debt when it comes due. 

1. What is the present worth of $440, due 2 yr. hence, at 
5%? 

Solution. — Since the amount of $1 for 2 yr. at 5% is $1.10, the 
present worth of $440 is as many dollars as $1.10 is contained times in 
$440, or $400. 

2. What is the present worth of $420, due 4 yr. hence, at 



3. What is the present worth of $26, due 5 yr. hence, at 

6%? 



126 MENTAL ARITHMETIC. 

4. What is the present worth of $600, due 4 years hence, 
at 5% ? 

5. What is the present worth of $360, due 3^ yr. hence, 
at 6%? 

6. What is the present worth of $530, due 18 months 
hence, at 4% ? 

7. The amount due on a mortgage which has been on in- 
terest for 8 years at 5% is $4900; what is the face of the 
mortgage ? 

8. If I buy a bill of paper amounting to $203, payment 
to be made in 3 months, what amount of cash will pay the 
bill now, interest at 6% ? 



TRADE DISCOUNT. 

Manufacturers, and sometimes wholesale merchants, have 
a fixed price on their goods, known as their list price. From 
this they make discounts varying as the market varies. Such 
discount is known as the trade discount^ and the amount paid, 
as the net price. Frequently several discounts are given ; as, 
10% off and 5% for cash ; or 30% and 10% off. 

In all such cases the discounts are deducted separately from 
the remainder left after each preceding discount has been de- 
ducted. Thus, in goods sold at $40 with 10% off and 5% 
for cash, 10% of $40, or $4, is first deducted, and then 5% 
from the $36 remaining. 

1. Find the net price of goods sold at $25, discount 20 
and 10. 

Solution.— 20% of $25 is $5; $25— $5 equals $20; 10% of $20 is 

$2; $20 — $2 = $18. 

2. Sold a bill of goods for $60, discount 20% and 5% off 
for cash ; what was the net price ? 



BANK DISCOUNT. 127 

3. If I buy books at $10 a dozen, discount being 40 and 
10, what is the net price? 

4. The list price of goods is $1.50 ; what is the net price 
if the discount is 10 and 20 ? 

5. Which is the better for the purchaser, a discount of 10 
and 20, or a discount of 20 and 10 ? 

6. The list price of shoes is $40 a dozen, and the discount 
is 25 and 10 ; what do they cost a pair? 

7. The list price of goods on 3 months' time is $4, the dis- 
count is 10% and 5% for cash ; what is the net price? 

8. What is the net price of goods listed at $10, discount 
40, 25, and 10? 



BANK DISCOUNT. 

Bank Discount is the allowance made to a bank for the 
payment of a note before it comes due. 

Bank discount is the interest on the face of the note for 
the time it has to run, including three days of grace. 

1. What is the bank discount of $120 for 60 days at 6% ? 
SuG.— The time is 63 days; the interest of $120 for 63 days at 6% 

is 11.26. 

Find the bank discount of — 

2. $100 for 60 da. at 6%. 10. $720 for 90 da. at 6%. 

3. $120 for 30 da. at 6%. 11. $200 for 60 da. at 5%. 

4. $200 for 90 da. at 8%. 12. $500 for 90 da. at 6%. 

5. $600 for 30 da. at 10%. 13. $300 for 95 da. at 6%. 

6. $140 for 60 da. at 6%. 14. $800 for 60 da. at 5%. 

7. $125 for 60 da. at 6%. 15. $500 for 30 da. at 6%. 

8. $700 for 60 da. at 6%. 16. $150 for 45 da. at 6%. 

9. $300 for 45 da. at 6%. 17. $120.30 for 90 da. at 6%. 



128 MENTAL ARITHMETIC. 



PROPORTIONAL PARTS. 

1. Two men hire a pasture for $60: the first pastures 4 
cows and the second 6 cows ; how much should each pay ? 

Solution. — Since the first pastured 4 cows and the second 6, they 
together pastured 10 cows. If the pasturage of 10 cows cost $60, the 
pasturage of 1 cow cost y\j of |60, or $6 ; hence the pasturage of 4 cows 
cost 4 times $6, or $24, and the pasturage of 6 cows, 6 times $6, or $36. 

2. If two men hire a horse for $25, the first using it 10 
days and the second 1 5 days, how much should each pay ? 

3. Three boys bought 60 apples, the first paying 3 cents, 
the second 4 cents, and the third 5 cents ; how many apples 
should each receive ? 

4. The pasturage of 5 cows for 4 weeks and 10 sheep for 
4 weeks cost $30 ; how much does the pasturage of each cost, 
if the pasturage of a cow cost 4 times as much as the pastur- 
age of a sheep ? 

5. Smith and Martin undertake to do a piece of work for 
$100 : Smith sends 5 men for 6 days, and Martin 4 men for 5 
days ; how much should each receive ? 

6. 3 men, A, B, and C, undertake to do a piece of work for 
$348 : A with 3 assistants works 10 days ; B with 5 assistants 
works 6 days ; and C with 9 assistants works 4 days ; how 
much should each receive ? 

7. A furnishes 10 men, and B 16 boys, to do a piece of 
work for $90 ; how much should each receive, if the work of 
a man is worth twice that of a boy ? 

8. Three men hire a pasture for $124. C puts in 4 horses ; 
D 10 cows ; and E 30 sheep ; how much should each pay if 
a cow eats twice as much as a sheep, and a horse three times 
as much as a sheep ? 

9. A puts into a pasture 20 sheep for 5 weeks ; B, 10 cows 
for 4 weeks ; and C, 5 horses for 6 weeks ; how nmch should 



PEOPORTIONAL PARTS. 129 

each pay if a cow eat twice as much as a sheep, and a horse 
twice as much as a cow, the rent of the pasture being $60? 



10. Divide the number 180 into three such parts as shall 
be to one another as 3, 4, and 5. 

Solution. — Since the parts are to one another as 3, 4, and 5, we 
divide the number 180 into 3 + 4 + 5, or 12, equal parts: 1 of these 
parts is j^ of 180, or 15 ; 3 parts are 3 times 15, or 45 ; 4 parts are 4 
times 15, or 60 ; and 5 parts are 5 times 15, or 75. 

11. Divide 60 into two parts that are to each other as 5 
and 7. 

12. The sum of three numbers is 200, and the numbers are 
to one another as 3, 2, and 5 ; what are the numbers ? 

13. Divide 650 in the proportion of ^, |, and |. 

14. Three men caught 120 fish, their proportions being as 
2, 1^, and ^ ; how many did each catch ? 



15. Mary and Susan have $45: Mary has $5 more than 
Susan ; how much money has each ? 

Solution. — Susan's number plus $5 equals Mary's, which, added to 
Susan's, equals twice Susan's, plus $5, or $45; hence $45 minus $5, or 
$40, equals twice Susan's number. Susan's number is therefore J of 
$40, or $20, and Mary's is $20 plus $5, or $25. 

16. Divide the number 33 into two such parts that one 
shall be 5 greater than the other. 

17. A watch cost $50 more than a chain, and they together 
cost $110 ; how much did each cost? 

18. My coat cost $20 more than my boots, and together 
they cost $40; how much did each cost? 

19. A hat cost $3 more than a pair of shoes, and a shawl 
$8 more than the hat ; they together cost $25 ; what was the 
cost of each? 



ISO MENTAL AHITHMETIC. 

20. James and John gathered 60 melons ; James gathered 
3 times as many as John ; how many did each gather ? 

Solution. — Since James gathered 3 times as many as John, 3 times 
John's number equals James' number, which, added to John's, is 4 
times John's, or both. If 4 times John's number is 60, his number is 
J of 60, or 15, and James' number is 3 times 15, or 45. 

21. Two boys have 30 marbles : the first has twice as many 
as the second ; how many has each ? 

22. Mary and Ella gathered 60 roses: Ella gathered 2 
times as many as Mary ; how many did each gather ? 

23. The ages of two men are 45 years : one is ^ as old as 
the other ; how old is each ? 

24. If Charles and Simon gather 24 quarts of berries, and 
Charles gathers If times as many as Simon, how many quarts 
does each gather ? 

25. Two men have $3400 : | of what the first has equals | 
of what the second has ; how much money has each ? 

26. A house, a farm, and a store cost $12,000 : the farm 
cost twice as much as the house, and the store 3 times as 
much as the house; how much did each cost? 

27. A man bought three properties : the second cost twice 
as much as the first, and the third 3 times as much as the 
second ; what was the cost of each, if they together cost 
$13,500? 

28. Three times Henry's money equals 4 times Arthur's : 
they together have $140 ; how much has each ? 

29. Two boys sold 90 tickets ; how many did each sell if 4 
times what the first sold equals 5 times what the second sold ? 

30. The sum of two fractions is ly^r : | of the first equals 
f of the second ; what are the fractions ? 

31. Bought a hen, a duck, and a turkey: the duck cost 15 
cents less than the hen, and the turkey 3 times as much as 
the duck ; what was the price of each, if they together cost 
$1.90? 



PARTNERSHIP. 131 



PARTNERSHIP. 



1. Two men in partnership gain S600. B owns | of the 
stock, lacking $100, and his gain is $400 ; how much stock 
has each ? 

Solution. — If A had owned | of the stock, his gain would have been 
f of $600, or $450; but he gained only $400, therefore the $100 would 
have gained the difference between $450 and $400, or $50. If it re- 
quires $100 to gain $50, it requires $2 to gain $1, and to gain $600 it 
requires 600 times $2, or $1200; hence A had f of $1200, or $900, less 
$100, which equals $800; and B, $1200 minus $800, or $400. 

2. Mr. Smith and Mr. Handy enter into partnership and 
gain S300. Mr. Smith owns | of the stock, less $200, and his 
share of the gain is $60 ; how much stock does each own ? 

3. Mr. A and Mr. B gain in partnership $320. Mr. A 
owns I of the stock, + $60, and his share of the gain is 
$110; what amount of stock has each? 

4. C and D bought a store, and in selling it cleared $1500. 
C owned ^ of it, less $600, and his share of the gain was 
$600 ; what amount did each pay ? 

5. Charles and Henry agree to plow some land for $60. 
Charles plowed ^ of the land and 10 acres more, and received 
$40 ; how much did each plow ? 

6. If a man builds ^ of a wall and 6 rods more and re- 
ceives $48, how long is the wall, if the total cost of building 
it is $120 ? 

THE METRIC SYSTEM. 

The Metric System is a decimal system of weights and meas- 
ures having the meter for its base or unit. 

The system was first adopted in France in 1795, and it is 
now used in France, Germany, Spain, Portugal, Belgium, 
Greece, Mexico, Brazil, and most of the other states of South 
America. 



132 MENTAL ARITHMETIC. 

Its use is authorized by law also in Great Britain and the 
United States. The system is used chiefly by scientific men, as 
being the simplest system of weights and measures in existence. 

The principal metric units are — 

•1. The Meter, for lengths. 

2. The Are, or square meter, for surfaces. 

3. The Stere, or cubic meter, for volumes. 

4. The Liter (lee'ter), or cubic decimeter, for capacities. 

5. The Gram, for weights. 

The Meter, the primary unit of the system, is 39yy^ inches. 

The names of the denominations lower than a unit are in- 
dicated by prefixing the Latin words milli (yoVo)> ^^^^* (tJ^)> 
and ded (j\) to the unit ; thus, millimeters, centimeters, etc. 

The names of the denominations greater or higher than a 
unit are formed by prefixing the Greek terms deca (10), hecto 
(100), kilo (1000), and myria (10,000) to the unit ; thus, deca- 
grams, hectograms, etc. 

Metric numbers are written as decimals, with the decimal 
point after the unit. Thus, 15 meters and 5 centimeters is 
written 15.05 m. This may be read 15 and y^^ meters, or 
15 meters and 5 centimeters. 

The following are the tables of the metric system : 





1. 




Linear Measure. 




10 millimeters (mm.) 


= 1 centimeter 


(cm.) 


10 centimeters 


= 1 decimeter 


(dm.) 


10 decimeters 


— 1 meter 


(m.) 


10 meters 


— 1 decameter 


(Dm.) 


10 decameters 


= 1 hectometer 


(Hm.) 


10 hectometers 


= 1 kilometer 


(Km.) 


10 kilometers 


= 1 myriametei 


• (Mm.) 



THE METRIC SYSTEM. 133 

The Meter, a little less than 1 yd. 3| inches, is used in 
measuring short distances. 

The Kilometer, a little less than f of a mile, is used in 
measuring long distances. 

2. 

Surface Measure. 

100 sq. millimeters (mm.) = 1 sq. centimeter (sq. cm.) 

100 sq. centimeters = 1 sq. decimeter (sq. dm.) 

100 sq. decimeters = 1 sq. meter (sq. m ) = 1 centiare (ca.) 

100 sq. meters = 1 sq. decameter (sq. Dm.) = 1 are (a.) 

100 sq. decameters = 1 sq. hectometer (sq. Hm.)r=l hectare (Ha.) 

100 sq. hectometers = 1 sq. kilometer (sq. Km.) 



Also- 



100 centiares = 1 are. 
100 ares — 1 hectare. 



Land is measured by the Are, about 10| sq. feet, and the 
Hectare, about 2| acres. 

3. 

Cubic Measure. 

1000 cu. millimeters (cu. mm.) = 1 cu. centimeter (cu. cm.) 
1000 cu. centimeters = 1 cu. decimeter (cu. dm.) 

1000 cu. decimeters = 1 cu. meter = 1 stere. 

The cubic meter, or stere, used for measuring wood, em- 
bankments, etc., is about 1^ cu. yard, or .2759 of a cord. The 
following table also is used : 

10 decisteres (ds.) = 1 stere (st.) 
10 steres = 1 decastere (Ds.) 



134 MENTAL ARITHMETIC. 

Liquid or Dry Measure. 

The Liter is the unit of the measure of capacity. It is 
equal to a cube each of whose sides is j\ of a meter, and is 
equivalent to about l^^g liquid quarts or -^^ of a quart dry- 
measure. 

10 milliliters (ml.) - 1 centiliter (cl.) 



10 centiliters 


= 1 deciliter 


(di.) 




10 deciliters 


= 1 liter 


(!•) = 


1 cu. dm. 


10 liters 


= 1 decaliter 


(Dl.) 




10 decaliters 


= 1 hectoliter 


(HI.) 




10 hectoliters 


= 1 kiloliter 


(Kl.) 





Note. — 1 milliliter = 1 cu. cm. 

The Hectoliter is used in measuring liquids, grains, fruits, 
and roots in large quantities. It is equal to about 2| bu. 
A Kiloliter is a cubic meter or stere. 

5. 

Weight. 

The Gram, the unit of weight, is equal to a cube of dis- 
tilled water the edge of which is yj^ of a meter, equal to 
15.432 Troy grains in weight. 

10 milligrams (mg.) = 1 centigram (eg.) 

10 centigrams = 1 decigram (dg.) [of water. 

10 decigrams = 1 gram (g.) — Wt. of 1 cu. cm. 

10 grams , = 1 decagram (Dg.) 

10 decagrams = 1 hectogram (Hg.) [water. 

10 hectograms = 1 kilogram (Kg.) = 1 cu. dm. of 

10 kilograms = 1 myriagram (Mg.) 

10 myriagrams = 1 quintal. 

10 quintals = 1 touneau or ton == 1 cu. m. of water. 



PROBLEMS FOR ANALYSIS. 135 

The Gram, the Kilogram (Kilo), and the Ton are the 
units used in actual weighing, except in cases where jewelers, 
druggists, and others find it necessary to use the other units 
in weighing very small or very expensive articles. 

The Kilogram, or Kilo, is used in common trade. It is a 
little less than 2^ lb. avoirdupois. 

The Tomieau, or Ton, equal to 1000 kilos, is used for 
weighing heavy articles. It is equal to a little less than 
2205 lb. 

PROBLEMS FOR ANALYSIS. 

1. If I buy a certain number of hens at 40 cents, I shall 
have 60 cents remaining, but if I buy the same number at 
45 cents, I shall have only 10 cents remaining ; how many 
are there? 

Solution. — By the conditions of the problem each hen at the sec- 
ond price will cost 5 cents more than a hen at the first price, and the 
difference in the price of the whole number is 60 cents less 10 cents, or 
50 cents ; hence there are as many hens as 5 is contained times in 50, 
or 10. 

2. A girl bought apples at 1 cent each and had 12 cents 
remaining ; had she bought as many oranges at 3 cents each, 
she would have had no money remaining ; how many apples 
did she buy? 

3. Mr. Rogers bought sheep at $5 each and had $45 re- 
maining ; had he bought as many hogs at $7 each, he would 
have had only $5 remaining ; how many sheep did he buy ? 

4. A boy bought^ ducks at 35 cents each and had 30 cents 
remaining ; had he bought the same number of hens at 25 
cents each, he would have had 80 cents remaining; how 
many did he buy? 

5. A farmer, desirous of purchasing a certain number of 
cows, found that if he bought those at $30 each, he would 
have $40 remaining, but if he bought those at $40 a head, he 



136 MENTAL ARITHMETIC. 

would lack $20 of having money enough to pay for them ; 
how many did he wish to buy? 

6. James bought a number of oranges at 3 cents each and 
had 6 cents remaining ; had he bought the same number of 
bananas at 5 cents each, he would have lacked 4 cents to pay 
for them ; how many did he buy ? 

7. A gentleman gave to an equal number of boys and girls 
$1.80, giving to each boy 4 cents and to each girl 5 cents ; 
how many were there of each ? 

Note. — He gave to 1 boy and 1 girl, 9 cents ; hence there were as 
many of each as 9 is contained times in 180. 

8. I bought an equal number of oranges and bananas for 
49 cents, giving 4 cents for the oranges and 3 cents for the 
bananas ; how many did I buy of each ? 

9. I bought an equal number of pears, apples, and peaches 
for 60 cents, the pears costing 5 cents, the peaches 3 cents, 
and the apples 2 cents each ; how many of each did I get ? 

10. A gentleman distributed 90 cents among some poor 
children, giving to each child 6 cents : there were twice as 
many girls as boys ; how many were there of each ? 

11. Mr. B distributed 84 cents among some poor children, 
giving to each girl 5 cents and to each boy 4 cents ; how 
many were there of each if there were twice as many girls 
as boys? 

12. Samuel bought a number of oranges at 5 cents each, 
and three times as many apples at 2 cents each, spending for 
all 66 cents ; how many of each did he buy ? 

13. Three men agree to build a wall for $55 : A is to have 
$1^ a day, and B and C each $2 a day ; how long does it take 
to build the wall ? 

14. Mr. Brown and Mr. Smith agree to do a piece of work 
for $110, Brown receiving $2^ a day, and Smith receiving $2 
a day, but Smith works 1^ times as many days as Brown ; 
how many days did each work? 



PKOBLEMS FOR ANALYSIS. 137 

1. A miller paid $2S for 50 bushels of corn of two grades : 
for the first he paid 60 cents a bushel, and for the second 50 
cents a bushel ; how many bushels of each did he buy ? 

Solution. — Had all the corn been of the first grade, he would have 
paid 130, or $2 more than he did pay ; each bushel of the second grade 
cost 10 cents less than a bushel of the first grade, hence there were as 
many bushels of the second grade as 10 cents is contained times in $2, 
or 20 bushels. The number of bushels of the first grade was 50 — 20, 
or 30. 

2. A drover sold 40 sheep for S210 : for the best he received 
$6 each, and for the others $5 each ; how many were there of 
each? 

3. Bought 30 hens and ducks for $14 : the hens were worth 
50 cents each, and the ducks 40 cents each ; how many were 
there of each ? 

4. Paid a coal dealer $60 for 12 tons of coal: for stove 
coal I paid $5.25 a ton, and for nut coal $4.50 ; how many 
tons were there of each ? 

5. A gentleman paid $90 to 40 laborers for a day's work : 
the men received $2.50 each, and the boys $1.50 each ; how 
many of each were there ? 

6. Paid $9.80 for 12 bushels of grain : wheat at 90 cents a 
bushel and corn at 70 cents a bushel ; how many bushels were 
there of each ? 



7. A man agreed to work 40 days for $1.25 a day and board, 
but was to pay 75 cents a day for his board whenever he was 
idle ; at the end of the time he received $42 ; how many days 
was he idle ? 

Solution. — Had he worked every day, he would have received 40 
times $1.25, or $50; he therefore lost by idleness $50 less $42, or $8. 
His loss for a single day was his wages, $1.25, and his board, 75 cents, or 
$2 ; hence he was idle as many days as $2 is contained times in $8, or 
4 days. 



138 MENTAL ARITHMETIC. 

8. A mechanic agrees to work at the rate of $3 a day for 
30 days : at the end of the time he receives $75 ; how many 
days did he lose ? 

9. A laborer agrees to work 20 days at 75 cents a day and 
board, but he is to pay 50 cents a day boarding for every day 
he is idle: at the end of the time he receives $12.50 ; how 
many days does he work ? 

10. A man agrees to work 40 days for $100, but at the 
end of the time gets only $90 ; how many days was he idle ? 

11. An engineer agrees to work 25 days for $100, but for- 
feits $1 for every day he is idle : at the end of the time he 
receives $85 ; how many days does he work ? 

12. A clerk who receives $5 a day for his work pays $7 a 
week for his boarding : at the end of 6 weeks he has saved 
$118 ; how many days did he lose ? 

13. A man receives $2 a day for his labor and pays 50 
cents a day for his board : at the end of 30 days he has saved 
$40 ; how many days did he work ? 



14. The head of a fish is 4 inches long, the tail is as long 
as the head, plus ^ of the length of the body, and the body 
is as long as the head and the tail both ; what is the length 
of the fish? 

Solution.— J of the length of the body, plus 4 inches, equals the 
length of the tail, which, added to the length of the head, equals J of 
the length of the body, plus 8 inches ; and this, by the condition of the 
problem, equals f , or the body. Since f of the length of the body 
equals J of the length of the body, plus 8 inches, | minus ^, or |, of 
the length of the body is 8 inches ; i of the length of the body is 4 
inches ; and |, or the body, is 3 times 4 inches, or 12 inches. The 
length of the fish is therefore 12 inches plus 4 inches plus 8 inches, 
or 24 inches. 

15. The foundation wall of a house is 3 ft. high ; the gable 
is as high as the foundation, plus ^ of the body of the house ; 



PROBLEMS FOR ANALYSIS. 139 

and the body is twice as high as the gable and the founda- 
tion ; what is the height of the house ? 

16. The head of a fish is 6 inches long, the tail is as long 
as the head, plus | of the body, and the body is as long as 
the head and the tail ; what is the length of the fish ? 

17. From A to B is 15 miles; from B to C is as far as 
from A to B, plus ^ the distance from C to D ; and from C 
to D is twice as far as from A to C ; how far from A to D if 
A, B, C, and D are in a straight line ? 

18. A has $10, B has as much as A, + | as much as C, and 
C has 3 times as much as A and B ; how much has each ? 

19. John earned $30, Henry earned as much as John and 
^ as much as Samuel, and Samuel earned twice as much as 
John and Henry; how much did each earn? 

20. George has 20 cents, Henry has as much as George 
and I as much as William, and William has as much as 
George and Henry together; how much has each? 



1. If a man can do a piece of work in 5 days, how much 
of it can he do in 1 day ? 

2. If a man do | of a piece of work in a day, how long 
will it take him to do | of it? How long to do the whole 
work ? 

3. If one boy can do a piece of work in 3 days, and another 
in 5 days, how much can each do in a day, and how much 
can both? 

4. George can do a piece of work in 4 days, Henry in 5 
days, and Isaac in 3 days ; how much can each do in a day, 
and how much can they all do? 

5. One man can do f of a piece of work in a day, and 
another | of the same work ; how much can they together 
do in 1 day, and how long will it take them to do the work 
together ? 



140 MENTAL ARITHMETIC. 

6. Arthur can do a piece of work in 1 J days, and Peter in 
21 days ; how much can each do in 1 day, and how much can 
they together do ? 

7. Two boys can dig a ditch in 3 days ; the first can dig it 
in 5 days ; how long will it require the second ? 

8. A boy can earn a dollar in | of a day ; how much can 
he earn in a day ? 

9. Henry can earn a dollar in | of a day, and Simon in | 
of a day ; how much can they together earn in a day? 

10. If John can earn a dollar in f of a day, and Calvin 
in f of a day, how long will it require each of them to 
earn $10? 

11. If James can earn a dollar in | of a day, and Frank 
in* I" of a day, how long will it take them, working together, 
to earn $8 ? 

12. A can do a piece of work in 4 days, B in 5 days, 
and C in 6 days; in what time can they together do the 
work? 

13. If George and Luther can plow a piece of land in 3 
days, and George alone can plow it in 5 days, how long will 
it take Luther alone to plow it ? 

14. Mr. Thompson can build a wall in 20 days, but he and 
Mr. Robinson together can build it in 12 days; how long 
would it take Mr. Robinson alone to build it? 

15. If one pipe can fill a cistern in 8 hours, and another in 
12 hours, how long will it take if they work together ? 

16. A cistern has three pipes, two for filling and one for 
emptying it. One pipe can fill it in 6 hours, and the other in 
10 hours, while the third can empty it in 8 hours ; how long 
will it take to fill it if all the pipes are flowing at the same 
time? 

17. If a barrel of flour last two families 3 months, and it 
last the first family 5 months, how long would it last the 
second ? 



PROBLEMS FOR ANALYSIS. 141 

18. A, B, and C can do a piece of work in 3 days : A can 
do it in 8 days, and B in 9 days ; how long would it take C 
to do it? 

19. A, B, and C can do a piece of work in 4 days : A and 
B can do it in 6 days, and B and C in 7 days ; how long 
would it take each alone to do the work? 

20. D, E, and F can build a wall in 6 days : D and E can 
build it in 10 days, and E alone in 16 days ; how long would 
it take each to build it ? 

21. A and B can do a piece of work in 8 days, B and C 
in 10 days, and A and C in 9 days ; how long would it take 
each to do it, and how long all of them when they work 
together ? 

22. Jacob and Silas can mow a field in 4 days, and Silas 
alone can mow it in 6 days ; how long will it take Jacob 
to mow the remainder after they have worked together 3 
days? 

23. B and C can clear a piece of land in 3 weeks : B alone 
can clear it in 5 weeks ; how long will it require C to clear 
what remains after B has worked 3 weeks ? 

24. 2 men or 3 boys can do a piece of work in 10 days ; 
how long will it take 2 men and 3 boys to do the work? 

25. 2 men or 3 boys can do a piece of work in 8 days ; how 
long will it take 2 men and 2 boys to do it ? 

26» 2 men or 3 boys can do a piece of work in 6 days ; how 
long will it take 3 men and 2 boys ? 

27. John can earn a dollar in | of a day, and Henry in -J- 
of a day ; how long will it take them together to earn a dol- 
lar, and how many dollars can they earn in a day ? 

28. Howard can plant 40% of a field in a week, and James 
can plant 30% of it in a week ; how long will it require them 
together to plant the field ? 



142 MENTAL ARITHMETIC. 

1. Mr. A bought a number of turkeys for $50, but 10 of 
thera having been stolen, he sold ^ of the remainder at cost 
and received $12.50; how many did he buy? 

Solution.— Since J of the remainder cost $12.50, |, or the remain- 
der, cost 3 times $12.50, or $37.50; and since the whole number cost 
$50, the ten that were stolen cost $50 less $37.50, or $12.50: if 10 tur- 
keys cost $12.50, 1 cost iV of $12.60, or $1.25; hence Mr. A bought as 
many as $1.25 is contained times in $50, or 40 turkeys. 

2. Mary purchased a quantity of gingham for $3.20, but 
after using 4 yards of it she sold | of the remainder for cost 
and received $1.80 ; how many yards did she buy? 

3. A boy had some ducks worth $8 : he killed 4, and then 
sold j of the remainder for $2 ; how many had he at first? 

4. Mr. Jones bought a number of geese for a certain sum : 
he lost 4, and then sold ^ of the remainder at cost and re- 
ceived $12, which was $18 less than all cost; how many had 
he? 

5. A lady had a number of chickens : she killed 10, then 
sold f of the remainder for $10, which was $20 less than the 
value of her flock at first ; how many had she at first ? 

6. A farmer, having purchased a number of sheep, sold 8 
to his neighbor, and then ^ of the remainder to a butcher at 
cost for $20, which was $80 less than the flock cost ; how 
many did he buy ? 

7. Mr. Jackson, having purchased some cows, sold 2 at 
cost, and sold J of the remainder at $20 more than cost, re- 
ceiving $140, which was $160 less than the lot cost; how 
many did he buy? 

8. Samuel bought a number of pigs for $80 ; he killed 4, 
and then sold ^ of the remainder, lacking 2, at cost and re- 
ceived $24 ; how many did he buy ? 

9. Henry purchased a number of hens for $30: having 
killed 10, he sold f of the remainder and 2 more at cost and 
received $11; how many did he buy? 



PROBLEMS FOR ANALYSIS. 143 

10. Mr. Evans, having bought a number of sheep for $80, 
sold 4 to a neighbor, and then sold to a butcher | of the re- 
mainder and 2 more, at cost, for $56 less than the cost of all ; 
how many did he buy ? 

11. A farmer lost ^ of his hogs : had he sold the remainder 
at cost, he would have received $80 ; but, reserving 4, he sold 
I of the remainder at cost and received $20 ; how many had 
he at first ? 

12. Mr. Williams bought a certain number of calves, and 
sold 5 of them to a butcher : had he sold the remainder at 
cost, he would have received $40 ; but he reserved 2, and, sell- 
ing I of the remainder at cost, he received $24 ; how many 
had he? 

13. Mr. A's family ate j of his chickens : he then found 
that if he sold | of the remainder at cost, he would receive 
$12, but if he kept 20 and sold | of the remainder at cost, 
he would receive $6 ; how many had he ? 

14. Mr. Lewis lost | of his sheep : now, if he finds 2, | 
of what he then has are worth at cost price $40 ; but if he 
loses 3, f of the remainder are worth only $30 ; how many 
had he? 



1. Two boys have 80 marbles : | of A's number equals f 
of B's ; how many has each ? 

2. A pole 76 feet in length is so broken that | of the 
shorter piece equals f of the longer piece; how long is 
each ? 

3. Henry and James together earned $77 : | of what Henry 
earned, + $6, equals | of what James earned ; how much did 
each earn ? 

4. I of Michael's money + $12, equals f of Albert's money ; 
they together have $46 ; how much has each ? 

5. A coat and a vest cost $27 : | the cost of the coat, plus 
$2, equals the cost of the vest ; how much does each cost ? 



144 MENTAL ARITHMETIC. 

6. Mr. Jones bought a horse, a cow, and some hens for 
$208 : the cow cost \ as much as the horse, and the hens ^ 
as much as the cow ; how much did each cost ? 

7. A lady had 66 fowls: | of the number of ducks equaled 
the number of turkeys, and | of the number of ducks equaled 
the number of hens ; how many had she of each ? 

8. George, having 60 cents, lost f of it, and then found J 
as much as he had remaining ; how much had he then ? 

9. Samuel found 10 cents, then lost \ of what he had, 
and still had j^jj as much as at first ; how much had he at 
first? 

10. William, George, and James earned $72 : William 
earned I as much as George, and George earned ^ as much 
as James ; how much did each earn ? 

11. Cyrus has 72 cents, which is | as much as Robert has ; 
if each give the other ^ his money, how much will each then 
have? 

12. Five times a number equals 6 times | of the same 
number, + 5 ; what is the number ? 

13. Four times a number equals 3 times the same number 
plus 5 ; what is the number ? 

14. Six times a number equals 4 times f of the same num- 
ber increased by 8 ; what is the number? 

15. A house and a lot cost $3000 : the lot cost I as much 
as the house ; how much did each cost? 

16. The cost of a house and a farm was $7800 : the house 
cost f as much as the farm, + $800 ; what was the cost of 
each? 

17. A man spent | of his money, then earned | as much as 
he spent, and had $27 ; how much had he at first ? 

18. Henry earned $40, then spent J of his money, and still 
had $50 ; how much had he at first ? 

19. A man, being asked his age, said, 10 years ago I was ^ 
as old as I am now : how old was he ? 



PBOBLEMS IB^OR ANALYSIS. l45 

20. Henry is f as old as his father, and the sum of their 
ages is 70 years; how old is each? 

21. If to my age you add its |, its |, and 16 years, the 
sum will be 3 times my age; how old am I? 

22. Saul is 16 years older than John, and f of Saul's age 
equals | of John's ; how old is each ? 

23. Mr. Rambo and Mr. Kerr own two farms : Mr. Rambo 
owns I as much land as Mr. Kerr, and Mr. Kerr owns 25 
acres more than Mr. Rambo ; how many acres does each own ? 

24. Mr. S and Mr. T own a number of sheep: Mr. S's 
number is 1^ times Mr. T's number, and Mr. T's number is 
40 less than Mr. S's; how many has each? 



1. In a school of 50 pupils there are 3 boys to every 2 girls ; 
how many boys must leave that there may be an equal num- 
ber of boys and girls ? 

2. In the same school how many girls should be admitted 
that the number of boys and girls may be equal ? 

3. In a school of 45 pupils the number of boys is to the 
number of girls as 3 to 2 ; how many boys must leave that the 
number of boys may be to the number of girls as 2 to 3 ? 

4. In the same school how many girls must be admitted to 
make the proportion of girls to boys as 3 to 2 ? 

5. In a school of 40 pupils, if the proportion of boys to 
girls is as 5 to 3, how many girls must be admitted to make 
the proportion of boys to girls as 5 to 6 ? 

6. In an alloy of 40 pounds there are 30 pounds of copper 
and 10 pounds of tin ; how much tin must be added that there 
may be 2 pounds of tin to every 3 of copper ? 

7. In an alloy of copper and silver there are 4 pounds of 
copper and 6 of silver ; how much copper must be added 
that there may be 2 pounds of copper to 1 of silver? 

8. A pole 124 feet in length was broken so that | of the 

10 



146 MENTAL ARITHMETIC. 

longer piece equaled § of the shorter ; how much must be cut 
from the longer that y% of its length may be | of the shorter ? 

9. If an alloy contain 30 oz. of gold and 2 oz. of silver, 
how much silver must be added that 24 oz. of the alloy may 
contain 20 oz. of gold ? 

SuG. — Since 24 oz. of the alloy contain 4 oz. of silver, 30 oz. will 
contain 1} times 4 oz. of silver, or 5 oz. of silver; there must there- 
fore be added 5 oz. minus 2 oz., or 3 oz., of silver. 

10. If an alloy contain 18 pounds of silver and 1 pound 
of copper, how much copper must be added that 10 pounds 
of the alloy may contain 9 parts silver and 1 part copper ? 

11. If alcohol is 90% strong, how much water may be 
added to a gallon so that the alcohol may remain 75% 
strong ? 

12. For every cow a farmer keeps he allows 2 acres of 
pasturage and 1 acre of corn ; how many cows can he keep 
on 12 acres? 

SuG. — Each cow requires 3 acres. 

13. If a farmer can pasture 2 cows on 4 acres, and 2 cows 
will eat the produce of 1 acre of roots, how many cows can 
he keep on 15 acres? 

14. If a man allow 2 acres of roots for every 5 cows he 
keeps and 4 acres of pasturage for every 2 cows, how many 
cows can he keep on 36 acres, and how much land must he 
have in pasture, and how much in roots? 

15. If a farmer allow one acre of corn for every 6 sheep 
and an acre of grass for every 4 sheep, how many sheep can 
he keep on 20 acres? 

16. Mr. A has 36 sheep: he allows 1 acre of corn for 
every 6 sheep and 1 acre of grass for every 4 sheep ; how 
many acres does he use? 

17. A farmer allows 6 acres of pasturage for every 3 cows 
and 1 acre of roots for every 3 cows ; he also allows 1 acre 



PROBLEMS FOR ANALYSIS. 147 

of corn for every 6 sheep and 2 acres of grass for every 8 
sheep; how many acres does it require to keep 12 cows and 
24 sheep? 

18. How far may a person ride in the cars at the rate of 
20 miles an hour and return by carriage at the rate of 8 miles 
an hour, if he is gone 7 hours ? 

Solution. — If he goes 20 miles an hour, he travels 1 mile in -^^ of 
an hour ; if he returns at the rate of 8 miles an hour, he travels 1 
mile in J of an hour ; hence to go one mile and return requires 2V H~i> 
or -^Qj of an hour. He therefore can go and return as many miles as 
5V is contained times in 7, or 40 miles. 

19. How far may a person ride in a coach going 6 miles 
an hour and be gone 9 hours, returning on foot at the rate of 
3 miles an hour ? 

20. I drive to the city at the rate of 6 miles an hour and 
return by rail at the rate of 30 miles an hour ; how far is it 
if I traveled the distance in 6 hours ? 

21. If I travel to the city by steamer at the rate of 15 
miles an hour and return by rail at the rate of 30 miles an 
hour, how far is it if I spend 2h hours in the city and return 
in 8^ hours ? 

22. A boat that sails at the rate of 10 miles an hour goes 
down a stream whose current is 2 miles an hour ; it returns 
in 5 hours ; how far did it travel ? 

23. A steamer goes up a river at the rate of 12 miles an 
hour and returns at the rate of 16 miles an hour ; it is gone 7 
hours ; how far did it travel, and what is the rate of the current? 



24. A dinner was engaged for 20 persons, but 4 were 
absent, whereby the expense of each of the others was in- 
creased 50 cents ; what was the total cost of the dinner ? 

Solution. — Since the expense of each of the remaining 16 was in- 
creased $1, the total increased expense was 16 times $|, or $8, which 



148 MENTAL ARITHMETIC. 

should have been paid by the 4 absentees : the expense to each was 
therefore i of $8, or |2, and to the 20, 20 times $2, or $40. 

25. Eight men hire a coach, but by taking in 4 others the 
expense of each was reduced 25 cents ; how much did they 
pay for the coach ? 

26. Four men buy a store, but by taking in an equal part- 
ner the amount to be paid by each was reduced $2000 ; what 
was the cost of the store ? 

27. Six men buy a boat, but afterward take in 2 partners, 
and thereby reduce the amount paid by each $5000 ; what 
was the amount paid for the boat ? 

28. An excursion party of 50 hire a boat, but 10 having 
failed to pay, the expense of each of the others was increased 
50 cents ; what was the charge for the boat ? 



29. Three boys. A, B, and C, camped together : A furnished 
4 loaves of bread, and B 5 loaves, while C paid 30 cents ; 
how should this money be divided ? 

30. Silas, Wilson, and Milton lunched together: Silas 
furnished 3 eggs, and Milton 6, while Wilson paid 10 oents ; 
to whom should the money be paid ? 

31. Four men camped together : A furnished 4 loaves of 
bread, B 5 loaves, and C 7 loaves, while D contributed 40 
cents ; how should the money be divided ? 



1. What is the time of day if I of the time past midnight 
equals the time to noon ? 

Solution. — Since | of the time past midnight equals the time to 
noon, f , or the time past midnight, plus I of the time past midnight, 
or the time to noon, equals f of the time past midnight, or the time 
from midnight to noon, 12 hours. If f of the time past midnight 
equals 12 hours, } of the time equals J of 12 hours, or 3 hours, and f, 
or the time past midnight, equals 3 times 3 hours, or 9 hours ; hence it 
is 9 o'clock A. M. 



PROBLEMS FOR ANALYSIS. 149 

Note. — It will be helpful to the pupil first to draw a diagram in 

solving these questions, as follows, in which P represents the present 

time: 

M # P 4 N 



2. What is the time of day if the time past midnight equals 
the time to noon ? 

3. What is the time of day if the time past noon equals ^ 
of the time to midnight ? 

4. What is the time of day if ^ the time past midnight 
equals the time to noon ? 

5. What is the time of day if ^ the time past midnight 
equals the time to noon ? ' 

6. What is the time of day if ^ the time past midnight 
equals the time to midnight again? 

7. What is the time of day if | of the time past noon 
equals the time to noon again ? 

8. What is the time of day if J the time past noon equals 
the time to midnight ? 

9. What is the time of day if ^ the time past midnight is 
the time past noon ? 

10. What is the time of day if | of the time to midnight 
is the time past noon ? 

11. What is the time of day if I the time past midnight 
equals the time past noon? 



12. How many more spaces does the minute-hand of a 
clock pass over in an hour, than does the hour-hand? 

13. How many spaces does the minute-hand gain on the 
hour-hand in 1 hour? In 2 hours? In 4 hours? 

14. How long does it take the minute-hand to gain one 
space on the hour-hand? 3 spaces? 6 spaces? 

15. At what time after one o'clock are the hour-hand and 
the minute-hand together ? 

Remark. — The minute-hand gains one space. 



150 MENTAL ARITHMETIC. 

16. At what time between 3 and 4 o'clock are the hour- 
hand and the minute-hand together? 

17. At what time between 4 and 5 o'clock do the hour- 
hand and the minute-hand point in opposite directions? 

18. At what time after 6 o'clock are the hour-hand and 
the minute-hand one space apart? 

19. It is after 4 o'clock, and the hour-hand and the minute- 
hand of a watch form a right angle ; what time is it ? 

20. It is between 1 and 2 o'clock ; what is the time if the 
hour-hand and the minute-hand of a watch form a right 
angle ? 

21. When f of the time to midnight equals the time past 
9 o'clock, what is the hour? 

22. When | of the time past 10 o'clock A. M. equals | of 
the time to 6 o'clock P. M., what is the time ? 

23. ^ of the time to 8 P. M. equals the time to noon ; what 
is the time ? 

1. A is 40 yards ahead of B ; if B runs 5 yards while A 
runs 4, how many yards must B run to overtake A? 

2. Henry is 20 feet ahead of Frank, but Frank runs 5 feet 
while Henry runs 2 ; how many feet will Frank run to over- 
take Henry ? 

3. A rabbit is 60 yards ahead of a hound, but the hound 
runs 5 yards while the rabbit runs 4 ; how far must the hound 
run to catch the rabbit ? 

4. A hound attempts to catch a rabbit which is 40 leaps 
ahead ; the rabbit takes 3 leaps while the hound takes 2, but 1 
of the hound's leaps equals 2 of the rabbit's ; how many leaps 
will each take before the hound catches the rabbit ? 

5. A thief is 30 steps before an officer, and takes 6 steps 
while the officer takes 5, but 5 of the officer's steps equal 7 
of the thief's; how far will each run before the thief is 
caught ? 



PROBLEMS FOR ANALYSIS. 151 

6. A man ran 160 yards to catch a boy ; -| of the distance 
the boy ran equals the distance he was ahead at the start ; 
how far was he ahead at the start? 

7. William takes 20 steps to overtake Edgar ; how far was 
Edgar ahead when they started if William takes 4 steps while 
Edgar takes 5, and 4 of William's equal 6 of Edgar's ? 

8. James and Nelson are 100 yards apart and approach 
each other, James walking 3 yards while Nelson walks 2 ; 
how far will each travel before they meet? 

9. Rufus and Edwin are 110 yards apart and approach 
each other; how far will each travel if Rufus takes 2 steps 
while Edwin takes 3, and 4 of Rufus' steps equal 5 of 
Edwin's ? 

10. Robert and Edward are 220 of Edward's steps apart, 
and approach each other; how many steps will each take 
before they meet if 3 of Robert's steps equal 5 of Edward's 
and Robert takes 2 steps while Edward takes 4 ? 



11. The sum of two numbers is 15, and twice the first plus 
4 times the second equal 48 ; what are the numbers ? 

Suggestion. — Twice the first and twice the second is 30, hence twice 
the second is 48 — 30, or 18. 

12. The sum of two numbers is 20, and twice the first plus 
3 times the second is 52 ; what are the numbers ? 

13. A cord of wood and a ton of coal cost $8, and 3 cords of 
wood and 5 tons of coal cost $34 ; what was the cost of each ? 

14. The difference between two numbers is 4, and 4 times 
the first less 3 times the second is 18 ; what are the numbers ? 

15. The difference between two numbers is 5, and 5 times 
the first less 2 times the second is 34 ; what are the numbers ? 

16. A man by working 12 hours a day can do a piece of 
work in 6 days ; how many days will it require if he work 8 
hours a day ? 



162 MENTAL. ARITHMETIC. 

17. If a man complete a piece of work in 5J days by work- 
ing 8 hours a day, in how many days can he finish it by work- 
ing 10 hours a day? 

1. Albert is 14 years old and Lucy is 6 ; how soon will 
Albert be twice as old as Lucy? 

Solution. — At the required time twice Lucy's age will equal 
Albert's age ; then the difference of their ages is once Lucy's age ; 
but the difference in their ages is always 14 — 6, or 8, years ; hence at 
the required time Lucy's age is 8 years. She k now 6 years old, hence 
in 8 — 6, or 2, years Albert will be twice as old as Lucy. 

2. Albert is 10 years old and his father is 40 ; how long 
before the father will be twice as old as Albert? 

3. Mary is 6 years old and James is 30 ; how long before 
James will be 3 times as old as Mary? 

4. Mr. Jones is twice as old as his son, who is 25 years 
old ; how long since he was 6 times as old ? 

5. B is 40 years old and his son 13 ; how long since B was 
4 times as old as his son ? 

6. Jennie is | as old as her father, who is 40 years old ; 
how long since she was ^ as old as her father? 

7. Harry is 4 times as old as Alice ; the difference in their 
ages is 21 years ; how long before he will be only twice as old 
as Alice ? 

8. Samuel is 3 times as old as Harry ; the sum of their 
ages is 40 ; how long ago was Samuel 5 times as old as 
Harry ? 

9. ^ of A's age is equal to | of B's, and the difference in 
their ages is 12 years; how soon will*! of A's age equal 
i of B's? 

10. I of John's age equals | of Henry's, and the difference 
of their ages is 3 years ; how long since John was twice as 
old as Henry? 



PROBLEMS FOR ANALYSIS. 153 

11. Philip is now 5 times as old as Susan, but in 8 years he 
will be only 3 times as old ; how old is each ? 

SoiiUTiON. — By the first condition of the problem the difference in 
their ages is 4 times Susan's age ; hence once her age is J of the differ- 
ence of their ages. In 8 years the difference in their ages is twice 
Susan's age, or her age is J of the difference, therefore 8 years is } — J, 
or :j, the difference of their ages, and f , or the difference of their ages, 
is 4 times 8 yeai-s, or 32 years. If 4 times Susan's age is 32 years, her 
age is J of 32 years, or 8 years ; and Philip's age is 5 times 8 years, or 
40 years. 

12. Henry is 6 times as old as Paul, but in 8 years he will 
be only twice as old as Paul ; how old is each ? 

13. A is 5 times as old as B, but in 3 years he will be only 
3 times as old as B ; how old is each ? 

14. Mrs. K is 3 times as old as her daughter, but in 10 
years she will be just twice as old as her daughter ; how old 
is each? 

15. When Mr. Porter married he was twice as old as his 
wife, but at the end of 20 years she was | as old as he ; how 
old was each when they married ? 

16. When Mr. Thomson married, his wife's age was |- of his, 
but after 6 years her age will be | of his ; how old was each ? 

17. Eighteen years ago Benjamin was ^ as old as his uncle ; 
now he is ^ as old ; how old is each ? 

18. Four years ago Ida's age was | her sister's age, but 12 
years hence it will be | her sister's age ; how old is each ? 



1. Mr. Smith bought a certain number of cattle for $120 : 
had he bought 2 additional at $10 less each, they would have 
cost him $160 ; how many cattle did he buy? 

Suggestion. — The additional two at $10 less each would have cost 
the difference between $160 and $120, or $40, hence each would have 
cost i of $40, or $20, which, increased by $10, is $30, the price of those 
purchased. Therefore there were as many bought as $30 is contained 
times in $120, or 4, 



154 MENTAL ARITHMETIC. 

2. A drover sold a certain number of sheep for $80 : had 
he sold 5 more at $1 less each, he would have received $100 
for the lot ; how many were there ? 

3. A farmer sold a number of hogs for $60 : had he sold 3 
additional at $2 more each, he would have received $96 ; how 
many did he sell ? 

4. Henry sold some hens for $4 : had he sold twice as many 
and 5 more, he would have received $10 ; how many did he 
sell? 

5. Mary sold some ducks for $1.50 : had she sold 3 times 
as many, less 2, she would have received $3.50 ; how many 
did she sell? 

6. Samuel bought a number of turkeys for $4.50 : had he 
bought 4 times as many, plus 2, they would have cost him 
$21 ; how many did he buy ? 

7. Mr. Jones bought some cows for $90: had he bought 
four times as many at $10 more each, they would have cost 
him $480; how many did he buy? 

8. George and William built a wall for $60 : George built 
3 times as many rods as William, less 12 rods, and received 
$36 ; how many rods did each build ? 

9. Two men agree to do a piece of work for $50 : the first 
works as many days as the second, less 3 days, and receives 
$22 ; how many days did each work ? 

10. Two men had an e^ual sum of money : had the first 
given the second 80 cents and the second given the first 20 
cents, the second would have had 4 times as much as the first; 
how much had each ? 

11. Henry and James agree to do a piece of work for $72 : 
I of the number of days Henry works, plus 4, equals | of the 
days James works, and Henry receives $32 ; how many days 
did each work ? 



PROBLEMS FOR ANALYSIS. 155 

1. A lady had two watches and a chain. The chain and 
the first watch were worth 3 times as much as the second 
watch ; and the chain and the second watch were worth as 
much as the first watch, the second watch being worth $40 ; 
what was the value of each ? 

Solution. — Since the first watch and the chain were worth 3 times 
as much as the second watch, they were worth 3 times $40, or $120, 
and the value of both watches and the chain was $120 plus $40, or 
$160. Since the value of the second watch and the chain was equal 
to the value of the first watch, the first watch was worth J of $160, or 
$80, and the chain was worth $120 — $80, or $40. 

2. A gentleman owned two horses and a carriage. The 
first horse and the carriage were worth twice as much as the 
second horse, and the carriage and the second horse were 
worth 1^ times as much as the first horse ; what was the 
value of each, if the second horse was worth $125? 

3. A man has a horse, a buggy, and a sleigh : the horse 
and the sleigh are worth twice as much as the buggy, and the 
buggy and the sleigh are worth as much as the horse ; what 
is the value of each, if the horse is worth $150? 

4. Mary has two cups and a cover that fits each : the first 
cup when covered weighs twice as much as the second cup, 
and the second cup when covered weighs ^ as much as the 
first cup ; what is the weight of each, if the first cup weighs 
10 ounces? 

5. SamueFs age is 16 years : his age and William's age are 
3 times Mary's age, and William's age added to Mary's age 
is 1-| times Samuel's; how old is each? 

6. Henry has 80 cents, and his money plus Irwin's is 4 
times Simon's ; Irwin's money plus Simon's is | of Henry's ; 
how much money has each? 



156 MENTAL ARITHMETIC. 

7. A boy lost 35 cents and then found ^ as much as he had 
remaining ; he then had ^ as much as at first ; how much had 
he at first ? 

Solution. — After finding ^ as much as he had remaining, he had 
f -f i> or h of what remained after losing 30 cents, which equals ^ of 
what he had at first ; hence ^ of the remainder equals ^ of J his first 
amount, or yV of what he had at first, and f equals ^^ of what he had 
at first ; the difference between |f and ^y, or ^i^j of what he had at first, 
is 35 cents, etc. 

8. John, having a certain sum of money, earned 50 dollars, 
and tlien spent $30, and still had 1^ times as much as at first ; 
how much had he at first? 

9. A man spent $30 and then earned | as much as he had 
remaining, and found that he had | as much as at first ; how 
much had he at first ? 

10. Simon, having a certain sum of money, borrowed 40 
cents, and then spending ^ of what he had, found there re- 
mained 1;^ times what he had at first; how much had he? 

11. A boy spent 15 cents and then earned 60 cents, when 
he found he had If times as much as at first ; how much had 
he at first ? 

12. Moore and Good lost $6000, and the next year gained 
J of the remainder of their capital, which was ^ of the original 
capital ; how much had each invested if Moore's capital was 
^ as much as Good's ? 

13. A and B lost 20 marbles, then found | as many as 
they had remaining, and then had | as many as at first ; how 
many had they each at first if A's number was twice B's? 

14. John and his brother, having earned a sum of money, 
spent $30, then earned | as much as they had remaining, and 
then had | as much as at first ; how much did each earn if 
John earned 1^ times as much as his brother ? 

15. If two men earn a certain sura of money in three 
weeks, then spend $15 each, and then earn | as much as 
they have remaining, and have I4 times as much as at first. 



PROBLEMS FOR ANALYSIS. 157 

how much does each earn in a week if the first earns 1| times 
as much as the second ? 



16. Henry went to a friend, borrowed as much money as 
he had, and gave away 6 cents ; he then went to a second 
friend, and, borrowing as much as he had, spent 16 cents, and 
had nothing remaining ; how much had he at first? 

Solution. — After borrowing the second time he had twice as much 
money as before he borrowed, or 16 cents ; therefore before borrowing, 
his money was | of 16 cents, or 8 cents, which he had after spending 6 
cents : before spending any he had 8 cents and 6 cents, or 14 cents, 
half of which, or 7 cents, he had before borrowing. 

17. A boy in one week earned as much money as he had 
and then spent $S ; he earned as much the second week as 
he had remaining at the end of the first week, and then had 
$20 ; how much had he at first ? 

18. George had a certain number of marbles. In play he 
won as many more, then he lost 10, and again won as many 
as he had remaining, and after losing 16 had none remain- 
ing ; how many had he at first ? 

19. A boy had a certain sum of money: he lost I of it, 
then earned as much as he had remaining, and, spending | 
of what he then had, there still remained 40 cents; how 
much had he at first? 

20. If, having a certain sum of money, I borrow an equal 
amount, then spend $6, and again borrow as much as I have 
remaining and spend $12, and still have $16 remaining, how 
much had I at first ? 



1. If a merchant buys goods at a discount of J-, what per 
cent, does he make by selling them at regular price ? 

2. If I buy goods at a discount of |, what per cent, profit 
do I make by selling them at regular price ? 



168 MENTAL AKITHMETIC. 

3. When wool that costs 25 cents a pound loses 10 per 
cent, in cleansing, how must I sell it per pound to make 20 
per cent, profit? 

4. If I buy goods at a discount of 20 and 10, what per 
cent, do I make by selling them at regular rates? 

5. A merchant bought books at a discount of 50% from 
retail price, and sold them at a discount of | from retail ; 
what per cent, did he make? 

6. A merchant bought iron at a discount of 40 and 10, 
and sold it at an advance of 25% ; what per cent, of the 
regular price did he receive? 

7. If I buy goods at a discount of 40 and 20, how must I 
sell them to gain 25% ? 

8. If a manufacturer gives a dealer a discount of 40 and 
25, and the dealer sells the goods at 60% of list price, what 
per cent, does he make? 

9. A merchant buys cloth worth $1.25 a yard, at a dis- 
count of 20 and 10 ; he sells it at regular price, $1.25, aud 
gives a discount of 10 per cent, for cash ; what profit does he 
make per yard ? 

10. If a merchant buys books worth $18 a dozen, at a dis- 
count of 40 and 10, and retails them at $1.20, allowing a dis- 
count of 5% for cash, what per cent, profit does he make ? 



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